# Tough Problem 1



## pgc01 (Jan 7, 2009)

Hi all

Many of us have fun creating solutions, imagining algorithms to solve difficult problems. Sometimes that's why we answer a question, because we think we'll have fun figuring out the solution.

I thought it would be a good idea if we posted, here in the lounge, complex problems to help us improve our knowledge in excel.

The idea is not to write a solution efficient, easy to read and to maintain. We do that in the excel forum. Here I thought we would try to push the formula/code capabilities to the limit. I'm talking about compact, ugly looking formulas, recursive algorithms, etc.

We will get solutions that we would not use in our work, but I believe it will help us to explore further the possibilities of the tools. Also the important is not the definitive solution, but the different ways to tackle the problems.

The problems should be complex enough to make even an experienced user feel some difficulty to solve them. 


I will post the first one. It requires some background in mathematics.

Tough Problem 1

Background:

As we all know, the prime numbers have a very irregular distribution. Sometimes we find pairs of primes that are only 2 units apart, like (17,19) or (2549,2551). They are called Twin Primes. Their distribution is also very erratic, you find 2 pairs between 820 and 830 (821,823) and (827,829) but not even one pair between 900 and 1000!

Problem:

How many pairs of Twin Primes are there between 5 and N, with N<=10000?

Ex.:

For N=20 we have

(3,5),(5,7),(11,13),(17,19) - 4 pairs

For N=100 we add the pairs

(29,31),(41,43),(59,61),(71,73) - total of 8 pairs

The solution must work in excel versions before 2007 (to avoid formulas with 25 nested parentheses).


Criteria for ranking of the solutions:

1 - less formulas is better
2 - cleverer, simpler solution is better

To the mvp's and experienced users: If you find this simple or already know a solution, please don't post it in the next hours to allow everyone to have some fun trying to figure it out.


Here is a link where you have the first 35 twin primes pairs:

http://en.wikipedia.org/wiki/Twin_prime

Don't forget, the idea is to have fun!!!

-----
P. S.
Do you think this makes sense?

Remark:
It may also happen that none of this makes sense and it's all the product of a disturbed mind.


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## barry houdini (Jan 7, 2009)

pgc01 said:


> How many pairs of Twin Primes are there between 5 and N, with N<=10000?


 
Hello pgc,

I don't have an answer (yet ) but I like the concept........

You say between 5 and N but list (3,5) as a pair, do you mean between 2 and N?

regards, barry


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## pgc01 (Jan 8, 2009)

barry houdini said:


> Hello pgc,
> 
> I don't have an answer (yet ) but I like the concept........
> 
> ...


 
Hi barry

No, I did mean between 5 and N, I should not have included the pair (3,5) in the examples.

Not many action here. I still think the concept is good. Maybe this was a bad choice for a first problem?

cheers


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## Domski (Jan 8, 2009)

> Do you think this makes sense?


 
It makes sense to me however I fear my frankly woeful maths skills will be below what is required.

I do like the concept though and will have a stab at it and look forward to seeing some of the solutions provided.

It's somewhat coincidental that you posted this as I came across Project Euler (http://projecteuler.net/) the other day and it got me thinking about similar challenges that I might be able to have a go at.

Best of luck folks,

Dom

PS If there's anyone in the class who I can copy off you can have half my dinner money


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## Jonmo1 (Jan 8, 2009)

Are you looking for only formulas, or will VBA functions be accepted?

=TwinPairs(5,100)


```
Public Function TwinPrimes(lownum As Double, highnum As Double)
Dim x As Long, i As Long
Dim MyPairs As String
x = 0
For i = lownum To highnum
    If isprime(i) Then
        If isprime(i + 2) Then
            x = x + 1
            MyPairs = MyPairs & " (" & i & "," & i + 2 & ")"
        End If
    End If
Next i
TwinPrimes = "There are " & x & " Twin Primes Between " & lownum & " And " & highnum & MyPairs
End Function
 
 
Public Function isprime(myval As Variant) As Boolean
Dim devisor As Double
Dim x As Double
isprime = True
For devisor = 2 To myval - 1
    x = myval / devisor
    If x = Int(x) Then 
        isprime = False
        Exit For
    End If
Next devisor
End Function
```
 

Also, I'm not sure about this.
Is there a built in IsPrime function, or did I reinvent the wheel?


Love the concept of the thread as well.


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## Jonmo1 (Jan 8, 2009)

I noticed an error in the logic.  It will give eroneous result if the Highest or Highest-1 Number is a prime.

Just change
For i = lownum To highnum
to
For i = lownum To highnum - 2

it will also be eroneous if lownum is 0-3, but the challenge stated to begin with 5, so to comply with that requirement, add a single line at the beginning

If lownum < 5 then lownum = 5


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## pgc01 (Jan 8, 2009)

Domski, thank you for the link, it's very interesting. I'll be spending some time there. As for the difficulty because of the required mathematical background, I agree. This may be the reason why there haven't been more posts so far (I hope).


Jonmo, thank you for posting, I'm expecting a formula solution. One of the sub-problems is to generate the list with the primes.


I'm posting a second problem for the vba fans. I think it's interesting to have both problems running at the same time, we know that many of us prefer formulas and other vba, so everyone is happy. Also it's very easy to understand and that may please more people.


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## cornflakegirl (Jan 9, 2009)

Jonmo - I love how simple that solution is!

I think you can replace
For devisor = 2 To myval - 1
with
For devisor = 2 To Int(Sqr(myval))

possibly slightly faster?

(PGC - apologies for hijacking with further VBA...)


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## pgc01 (Jan 9, 2009)

> PGC - apologies for hijacking with further VBA.


 
Hi Emma

On the contrary, thanks for posting. The better you understand the algorithms the easier you'll build the formulas.



> possibly slightly faster?



No, much, much faster. You should always use it when testing primes, and the bigger the number the more important that modification is.
Another thing you can do is to get rid of the even numbers.

This is another code to test the number


```
Function IsPrime1(lVal As Long) As Boolean
Dim l as Long
 
If (lVal = 1) Or (lVal > 2 And lVal Mod 2 = 0) Then Exit Function
 
For l = 3 To Int(Sqr(lVal)) Step 2
    If lVal Mod l = 0 Then Exit Function
Next l
IsPrime1 = True
End Function
```

If you test them, for example with a loop that tests the numbers from 2 to 50000 you see how faster it is.

P. S. Have you checked my other problem? Since you have a strong mathematical background you may find it intersting.


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## Jonmo1 (Jan 9, 2009)

> For devisor = 2 To Int(Sqr(myval))
> possibly slightly faster?


 
I don't understand why that would make it faster...
All that does is reduce the number of loops.

this bit does the same thing

```
If x = Int(x) Then 
        isprime = False
        Exit For
    End If
```
declares isprime false and exits the loop once the first evenly divisible number is found.


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## Jonmo1 (Jan 9, 2009)

Never mind, I get it now...

Any number larger than the sqare of the target number cannot be evenly devisible(without a an evenly divisible already previously found).

My original code would countinue needlessly testing if an evenly devisible had not been found prior to the square.


I like the skipping Evens too..


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## pgc01 (Jan 9, 2009)

Hi Jonmo

(deleted, you were quicker - I was posting a loop to illustrate what you just said)


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## Domski (Jan 9, 2009)

Okay, I've totally tried and failed!

I mentioned this to a friend of mine who's something of what I consider to be a math guru working for Nat Aus Bank and he's intrigued by both this challenge and that which I posted. If we get a chance next week we'll get together and use his math skills and a bit of my limited Excel knowledge and come back to you. In the meantime I'll just kick his *** at Wii Sports.

After a time maybe a seperate thread with some hints in the right direction rather than a straight solution could help some of us to achieve what we might otherwise have no hope of getting near.

Dom


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## Andrew Fergus (Jan 9, 2009)

This is pretty inefficient but it is purely formula based.

If the formula display isn't working, then here is each unique formula:
B4 : =COUNTIF(E:E,TRUE)
D5 (copy down) : =IF(AND(D4+2<$B$2,D4>3),D4+2,3)
E4 {array formula, copy down} : 
	
	
	
	
	
	



```
=(SUM(IF(MOD(D4,$D$2:D3)=0,1,0))=0) + (SUM(IF(MOD(D3,$D$2:D2)=0,1,0))=0) + (D3>=$B$1)=3
```
F4 (copy down) : =IF(E4,D3&", " &D4,"")


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align="right"><form name="formCb059465"><input onclick='window.clipboardData.setData("Text",document.formFb543420.sltNb142651.value);' value="Copy Formula" name="btCb290334" type="button"></form></td></tr></tbody></table></td></tr><tr><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(0, 0, 0);" colspan="7" bgcolor="white"><table border="0"><tbody><tr><form name="formFb543420"></form><td style="width: 60px;" align="middle" bgcolor="white"><select onchange="document.formFb543420.txbFb965317.value = document.formFb543420.sltNb142651.value" name="sltNb142651"><option value="=COUNTIF(E:E,TRUE)" selected="selected">B4</option><option value="{=(SUM(IF(MOD(D4,$D$2:D3)=0,1,0))=0)+(SUM(IF(MOD(D3,$D$2:D2)=0,1,0))=0)+(D3>=$B$1)=3}">E4</option><option value="=IF(E4,D3&&quot;, " &D4,"")">F4</option><option value="=IF(AND(D4+2<$B$2,D4>3),D4+2,3)">D5</option><option 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value="{=(SUM(IF(MOD(D9,$D$2:D8)=0,1,0))=0)+(SUM(IF(MOD(D8,$D$2:D7)=0,1,0))=0)+(D8>=$B$1)=3}">E9</option><option value="=IF(E9,D8&&quot;, " &D9,"")">F9</option><option value="=IF(AND(D9+2<$B$2,D9>3),D9+2,3)">D10</option><option value="{=(SUM(IF(MOD(D10,$D$2:D9)=0,1,0))=0)+(SUM(IF(MOD(D9,$D$2:D8)=0,1,0))=0)+(D9>=$B$1)=3}">E10</option><option value="=IF(E10,D9&&quot;, " &D10,"")">F10</option><option value="=IF(AND(D10+2<$B$2,D10>3),D10+2,3)">D11</option><option value="{=(SUM(IF(MOD(D11,$D$2:D10)=0,1,0))=0)+(SUM(IF(MOD(D10,$D$2:D9)=0,1,0))=0)+(D10>=$B$1)=3}">E11</option><option value="=IF(E11,D10&&quot;, " &D11,"")">F11</option><option value="=IF(AND(D11+2<$B$2,D11>3),D11+2,3)">D12</option><option value="{=(SUM(IF(MOD(D12,$D$2:D11)=0,1,0))=0)+(SUM(IF(MOD(D11,$D$2:D10)=0,1,0))=0)+(D11>=$B$1)=3}">E12</option><option value="=IF(E12,D11&&quot;, " &D12,"")">F12</option><option value="=IF(AND(D12+2<$B$2,D12>3),D12+2,3)">D13</option><option 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value="{=(SUM(IF(MOD(D21,$D$2:D20)=0,1,0))=0)+(SUM(IF(MOD(D20,$D$2:D19)=0,1,0))=0)+(D20>=$B$1)=3}">E21</option><option value="=IF(E21,D20&&quot;, " &D21,"")">F21</option><option value="=IF(AND(D21+2<$B$2,D21>3),D21+2,3)">D22</option><option value="{=(SUM(IF(MOD(D22,$D$2:D21)=0,1,0))=0)+(SUM(IF(MOD(D21,$D$2:D20)=0,1,0))=0)+(D21>=$B$1)=3}">E22</option><option value="=IF(E22,D21&&quot;, " &D22,"")">F22</option><option value="=IF(AND(D22+2<$B$2,D22>3),D22+2,3)">D23</option><option value="{=(SUM(IF(MOD(D23,$D$2:D22)=0,1,0))=0)+(SUM(IF(MOD(D22,$D$2:D21)=0,1,0))=0)+(D22>=$B$1)=3}">E23</option><option value="=IF(E23,D22&&quot;, " &D23,"")">F23</option><option value="=IF(AND(D23+2<$B$2,D23>3),D23+2,3)">D24</option><option value="{=(SUM(IF(MOD(D24,$D$2:D23)=0,1,0))=0)+(SUM(IF(MOD(D23,$D$2:D22)=0,1,0))=0)+(D23>=$B$1)=3}">E24</option><option value="=IF(E24,D23&&quot;, " &D24,"")">F24</option><option value="=IF(AND(D24+2<$B$2,D24>3),D24+2,3)">D25</option><option 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</td><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: black; font-family: menu; background-color: rgb(212, 208, 200);" align="middle"><center>A</center></td><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: black; font-family: menu; background-color: rgb(212, 208, 200);" align="middle"><center>B</center></td><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: black; font-family: menu; background-color: rgb(212, 208, 200);" align="middle"><center>C</center></td><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: black; font-family: menu; background-color: rgb(212, 208, 200);" align="middle"><center>D</center></td><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: black; font-family: menu; background-color: rgb(212, 208, 200);" align="middle"><center>E</center></td><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: black; font-family: menu; background-color: rgb(212, 208, 200);" align="middle"><center>F</center></td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>1</center></td><td style="border-style: solid; border-color: rgb(0, 0, 0) rgb(212, 208, 200) rgb(212, 208, 200) rgb(0, 0, 0); border-width: 0.5pt; font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">START</td><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(204, 255, 204); text-align: right;">5</td><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: left;">Number</td><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;">Pair?</td><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;">Pair</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>2</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">LIMIT</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(204, 255, 204); text-align: right;">100</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">2</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> </td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>3</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">3</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> </td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>4</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">PAIRS</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">7</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">5</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">FALSE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> 
</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>5</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">7</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">TRUE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;">5, 7</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>6</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">9</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">FALSE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> 
</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>7</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">11</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">FALSE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> 
</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>8</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">13</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">TRUE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;">11, 13</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>9</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">15</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">FALSE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> 
</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>10</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">17</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">FALSE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> 
</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>11</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">19</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">TRUE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;">17, 19</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>12</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">21</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">FALSE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> 
</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>13</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">23</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">FALSE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> 
</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>14</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">25</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">FALSE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> 
</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>15</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">27</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">FALSE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> 
</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>16</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">29</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">FALSE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> 
</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>17</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">31</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">TRUE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;">29, 31</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>18</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">33</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">FALSE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> 
</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>19</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">35</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">FALSE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> 
</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>20</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">37</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">FALSE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> 
</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>21</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">39</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">FALSE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> 
</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>22</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">41</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">FALSE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> 
</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>23</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">43</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">TRUE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;">41, 43</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>24</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">45</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">FALSE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> 
</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>25</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">47</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">FALSE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> 
</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>26</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">49</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">FALSE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> 
</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>27</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">51</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">FALSE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> 
</td></tr><tr><td style="border-top: 0.5pt solid rgb(0, 0, 0); border-left: 0.5pt solid rgb(0, 0, 0); font-size: 10pt; color: rgb(0, 0, 0); font-family: menu; background-color: rgb(212, 208, 200);" width="2%" align="middle"><center>28</center></td><td style="border-left: 0.5pt solid rgb(0, 0, 0); border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;"> </td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">53</td><td style="border-right: 0.5pt solid rgb(212, 208, 200); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: right;">FALSE</td><td style="border-right: 0.5pt solid rgb(0, 0, 0); border-bottom: 0.5pt solid rgb(212, 208, 200); font-size: 10pt; vertical-align: bottom; color: rgb(0, 0, 0); font-family: Arial; background-color: rgb(255, 255, 255); text-align: center;"> 
</td></tr><tr><td style="border-style: solid; border-color: rgb(128, 128, 128) rgb(0, 0, 0) rgb(0, 0, 0); border-width: 0.5pt; background-color: rgb(212, 208, 200);" colspan="7"><table valign="TOP" width="100%" align="left"><tbody><tr><td style="border-style: solid; border-color: rgb(128, 128, 128) rgb(0, 0, 0) rgb(0, 0, 0); border-width: 0.5pt; width: 120pt; background-color: rgb(255, 255, 255);" align="left">Sheet1</td><td> </td></tr></tbody></table></td></tr></tbody></table>
[HtmlMaker 2.42] To see the formula in the cells just click on the cells hyperlink or click the Name box
PLEASE DO NOT QUOTE THIS TABLE IMAGE ON SAME PAGE! OTHEWISE, ERROR OF JavaScript OCCUR.


</center>


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## barry houdini (Jan 9, 2009)

Here's a single formula to give the number of pairs, assuming N in A1

=SUM((FREQUENCY(ROW(INDIRECT("5:"&A1+3)),(MMULT((MOD(ROW(INDIRECT("5:"&A1)),TRANSPOSE(ROW(INDIRECT("2:"&A1))))=0)+0,ROW(INDIRECT("2:"&A1))^0)=1)*ROW(INDIRECT("5:"&A1)))=2)+0)

confirmed with CTRL+SHIFT+ENTER

It's very inefficient though and I couldn't get it to calculate for values of N > 5000 without my worksheet crashing 

I have a longer, slightly less costly formula but I'm still trying to improve it. I need a way to generate some smaller arrays (especially for the divisor in the MOD function) without compromising the accuracy......


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## pgc01 (Jan 12, 2009)

Hi Andrew

I tried your solution and it seems to work ok! It may require, however, several thousands of formulas. Can you, or anyone else, use that idea to come up with a more compact solution?

Cheers


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## pgc01 (Jan 12, 2009)

Great formula Barry!!!

If you can make work until 10000, it will be difficult for anyone to come up with a better one. I'm trying also, but my present formula is much (much,much) longer than yours, although it works till the 10000. I won't have much time today but I'll try to make it smaller.

Cheers


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## sous2817 (Jan 12, 2009)

This isn't my formula, but a solution I found posted at http://www.cpearson.com/excel/PrimeNumbers.aspx:



> The function below will return prime twin if the value in C15 is a prime and either C15-2 or C15+2 is a prime. If C15 is prime but neither C15-2 nor C15+2 is prime, the result is prime. If C15 is not a prime, the result is not prime. Like the formula above, this is an array formula so you must enter it with CTRL SHIFT ENTER. For readability here, the formula is split across several lines. In Excel, you will need to join these lines together.
> 
> =IF(OR(C15=1,C15=3),"prime twin",IF(AND((MOD(C15,ROW(INDIRECT("2:"&C15-1)))<>0)),
> IF(OR(AND((MOD(C15-2,ROW(INDIRECT("2:"&C15-3)))<>0)),
> ...



I thought I'd post it here to spur some discussion, as this is way outside of my abilities but makes for great reading!


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## Andrew Fergus (Jan 12, 2009)

Interesting solution but I think it contains an error in the very first part : OR(C15=1,C15=3),"prime twin"

1 is not actually a prime number!! 
Andrew


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## fairwinds (Jan 13, 2009)

barry houdini said:


> I need a way to generate some smaller arrays (especially for the divisor in the MOD function) without compromising the accuracy......


 

Hi,

This seems to work for me, using only array up to the square root of n.

{=SUM(--((FREQUENCY(((ROW(INDIRECT("5:"&A1-2))=TRANSPOSE(ROW(INDIRECT("2:"&INT(SQRT(A1))))))+(ROW(INDIRECT("7:"&A1))=TRANSPOSE(ROW(INDIRECT("2:"&INT(SQRT(A1))))))+MOD(ROW(INDIRECT("5:"&A1-2)),TRANSPOSE(ROW(INDIRECT("2:"&INT(SQRT(A1))))))*MOD(ROW(INDIRECT("7:"&A1)),TRANSPOSE(ROW(INDIRECT("2:"&INT(SQRT(A1))))))>0)*ROW(INDIRECT("5:"&A1-2)),ROW(INDIRECT("4:"&A1))))-INT(SQRT(A1))=-1))*(A1>6)-(A1=8)}


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## barry houdini (Jan 13, 2009)

Hello Fairwinds,<?xml:namespace prefix = o ns = "urn:schemas-microsoft-comfficeffice" /><o></o>
<o> </o>
Now that’s what I call a formula, I’ll need to go away and work out exactly how it works.<o></o>
<o> </o>
It prompted me to look again at my suggestion. I had realised that I could limit the MOD divisor range to the square root of N but that didn’t work within my original formula because my “primeness” test for each n of 1 to N was that it would have a single factor (not including 1), i.e. n itself. If I restricted it to SQRT(N) then some primes (those < SQRT(N)) would have no factors, and the others would have 1.<o></o>
<o> </o>
But I can change the formula slightly to take that into account (calculating whether each number needs 1 factor or 0 to be prime).<o></o>
<o> </o>
Having made that fix I realised that MMULT was still limiting the value of N to 5465 (although that restriction doesn’t apply in Excel 2007) so I changed the formula further to only consider odd numbers.<o></o>
<o> </o>
This formula, then, works on values of N up to 10926 (shouldn’t be a problem to go higher in Excel 2007 but I can’t access that version right now).

=SUM((FREQUENCY(ROW(INDIRECT("1:"&INT(A1/2+1.5)))*2+1,(MMULT((MOD(ROW(INDIRECT("2:"&INT(A1/2-0.5)))*2+1,TRANSPOSE(ROW(INDIRECT("2:"&INT(A1^(1/2))))))=0)+0,ROW(INDIRECT("2:"&INT(A1^(1/2))))^0)=(ROW(INDIRECT("2:"&INT(A1/2-0.5)))*2+1<=INT(A1^(1/2)))+0)*ROW(INDIRECT("2:"&INT(A1/2-0.5)))*2+1)=1)+0)

confirmed with CTRL+SHIFT+ENTER<o></o>


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## barry houdini (Jan 13, 2009)

.........Update.......

I used that last formula in Excel 2007 to calculate when N = 100,000 [the result was 1223] but any higher and I got some "out of resources" errors.


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## pgc01 (Jan 13, 2009)

Hi sous2817

Thanks for posting the formulas. THey seem to work ok, except for the 1 that, as Andrew already said, is not a prime. The second formula seems more efficient as only divisors up to the the square root of the dividend.


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## pgc01 (Jan 13, 2009)

Hi Fairwinds, Barry

I tried both solutions and they they both work ok, and both are great solutions!!! Barry's solution is more compact and faster to calculate.


I'll post my formula now:

=SUM(--(MMULT(--((3+2*ROW(INDIRECT("1:"&INT((A1-3)/2)))=TRANSPOSE(3+2*ROW(INDIRECT("1:"&INT((A1-3)/2)))))+(3+2*ROW(INDIRECT("1:"&INT((A1-3)/2)))=TRANSPOSE(3+2*ROW(INDIRECT("1:"&INT((A1-3)/2))))-2)),--(MMULT(1-(MOD(3+2*ROW(INDIRECT("1:"&INT((A1-3)/2))),TRANSPOSE(1+2*ROW(INDIRECT("1:"&ROUNDUP(A1^0.5/2,0)))))<>0)-(3+2*ROW(INDIRECT("1:"&INT((A1-3)/2)))=TRANSPOSE(1+2*ROW(INDIRECT("1:"&ROUNDUP(A1^0.5/2,0))))),--(1+2*ROW(INDIRECT("1:"&ROUNDUP(A1^0.5/2,0)))>0))=0))=2))

Both Fairwinds' and Barry's solutions are more efficient, faster and more compact than mine. I didn't use Frequency() and I guess that that's one of the reasons for my formula being slower. A strange thing that happened: I was working in excel 2000, to be sure that the solution would be compatible with versions before the excel 2007. When the solution was working in excel 2000, I tried it in excel 2007 and it only calculated until about 9500, then it says it's out of resouces!!! So, somehow the excel 2000 manages to be more efficient in the memory management than excel 2007??


I think we already have good solutions for this problem. I'll start thinking about the next one.

I think we got a very week participation in this problems and I think the reason was the difficulty of the problem. Although I did want these problems to be a challenge I should have posted as a first problem a simpler one. I'll try one more time, both a formula and a vba problems, and try to choose better the problems so that there's more discussion and participation.

Cheers


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## barry houdini (Jan 13, 2009)

Hello pgc,

Thanks for your comments. I thought it was an excellent challenge but, as you say, perhaps a little complex for mass participation.

If you are going to try more (and I think you should - if you run out of ideas I might have some of my own) then I suggest that you vary the level of difficulty, perhaps an easier one now followed by a more difficult one after?


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## pgc01 (Jan 14, 2009)

barry houdini said:


> Hello pgc,
> 
> Thanks for your comments. I thought it was an excellent challenge but, as you say, perhaps a little complex for mass participation.
> 
> If you are going to try more (and I think you should - if you run out of ideas I might have some of my own) then I suggest that you vary the level of difficulty, perhaps an easier one now followed by a more difficult one after?


 
Thank you Barry. That's a good idea, to alternate the difficulty of the problems, and I'm sure I'll take you up on your offer. 

Cheers


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