WR

[Xlambda]

WR: Word maze Results challenge. Please do see latest (26-Oct-21) YT video of MrExcel Solving Word Search With Excel (FMWC) - 2439
This is a compact set of functions that can solve all lookup scenarios.
Defined names: ar, sample array ( A1:T15 ) ; rw, nr. rows of sample array (15) (width is fixed to 20) ; cs, constant array values for "snake" lookup {-1,1, 20,-20} ; cd, constant array values for "diagonal" lookup {-21,-19,19,21}

WL(l)=LAMBDA(l,LET(x,IF(ar=l,1),y,IF(ar="?",1),z,IF(x+y,1),v,IF(z,SEQUENCE(rw,20)),SMALL(v,SEQUENCE(COUNT(v)))))
WA(a,c)=LAMBDA(a,c,LET(x,a-c,y,IF(x>0,x),UNIQUE(SMALL(y,SEQUENCE(COUNT(y ))))))
WB(a,b)=LAMBDA(a,b,FILTER(a,IFNA(XMATCH(a,b,,2),0)))
WR(w,c,[r],[i ])=LAMBDA(w,c,[r],[i ],LET(n,IF(i,i,LEN(w)),x,MID(w,n,1),y,MID(w,n-1,1),z,IF(AND(r=0),WA(WL(x),c),WA(r,c)),IF(n=1,r,WR(w,c,WB(z,WL(y )),n-1))))
Last one WR is !! recursive !! Define the names, then the functions and we are good to go. (Gaps in [i ] are to keep the text no to turn italics)
Will come back with the "concept" explanation asap.

=LAMBDA(w,c,[r],[i ],LET(n,IF(i,i,LEN(w)),x,MID(w,n,1),y,MID(w,n-1,1),z,
     IF(AND(r=0),WA(WL(x),c),WA(r,c)),IF(n=1,r,WR(w,c,WB(z,WL(y)),n-1))
    )
)

WORD MAZE.xlsx
	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R	S	T	U	V	W	X	Y	Z	AA	AB	AC	AD	AE	AF	AG	AH	AI	AJ	AK
1	Z	1	S	G	P	E	2	T	H	E	D	I	Z	B	A	N	T	S	N	S			defined names
2	S	P	?	A	K	I	W	K	1	F	H	V	Z	G	2	I	E	K	A	X			ar: A1:T15 (sample array)
3	?	I	T	Q	E	?	L	A	S	I	X	Q	J	N	S	X	4	T	J	K			rw: 15 (width is always 20)
4	L	E	N	J	F	X	A	4	T	I	K	N	S	B	X	1	D	J	C	P			cs: {-1,1,20,-20} (constant array for "snake" type lookup
5	G	D	L	3	3	D	M	V	L	T	G	K	?	C	U	3	O	Q	O	O			cd: {-21,-19,19,21} (constant array for "diagonal" type lookup
6	I	A	I	U	V	G	Y	L	U	?	I	E	P	3	W	G	K	B	T	P
7	4	A	P	A	U	4	C	Z	V	F	F	F	A	F	I	P	A	X	T	T			"snake" lookup						"diagonal" lookup
8	H	N	?	4	G	C	K	S	4	I	D	R	H	N	N	A	J	D	N	S
9	E	C	3	X	C	O	D	B	V	Q	O	Y	J	E	R	G	S	K	A	H			=WR("snake",cs)						=WR("diagonal",cd)
10	E	X	1	Z	?	P	N	I	3	O	W	C	C	N	2	U	R	E	G	J			3						82
11	E	O	N	M	E	V	K	A	B	E	P	F	L	Q	N	Z	?	E	W	3			18
12	K	A	N	?	A	I	B	K	L	G	2	4	?	N	?	Z	B	N	Q	E			20						=WR("advance",cd)
13	S	M	O	4	Y	F	B	B	R	E	?	D	N	E	1	A	D	J	M	4			160						67
14	C	F	X	4	3	T	Y	1	Z	J	N	I	W	U	Z	3	O	W	H	M			224
15	T	Y	D	2	Y	Z	S	U	D	V	V	F	4	O	Z	E	K	A	?	S			300
16
17	=SEQUENCE(rw,20)																						=WR("difficult",cs)
18	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20			151
19	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40
20	41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60			=WR("winner",cs)
21	61	62	63	64	65	66	67	68	69	70	71	72	73	74	75	76	77	78	79	80			115
22	81	82	83	84	85	86	87	88	89	90	91	92	93	94	95	96	97	98	99	100			273
23	101	102	103	104	105	106	107	108	109	110	111	112	113	114	115	116	117	118	119	120
24	121	122	123	124	125	126	127	128	129	130	131	132	133	134	135	136	137	138	139	140
25	141	142	143	144	145	146	147	148	149	150	151	152	153	154	155	156	157	158	159	160		Note: I left the results in "sequence numbers" representation on purpose,
26	161	162	163	164	165	166	167	168	169	170	171	172	173	174	175	176	177	178	179	180		to be much easier to visualize the results.
27	181	182	183	184	185	186	187	188	189	190	191	192	193	194	195	196	197	198	199	200		To get them to "range" representation we can use PR(a) (print results)
28	201	202	203	204	205	206	207	208	209	210	211	212	213	214	215	216	217	218	219	220		PR(a)=LAMBDA(a,LET(x,QUOTIENT(a-1,20)+1,y,MOD(a-1,20)+1,ADDRESS(x,y,4)))
29	221	222	223	224	225	226	227	228	229	230	231	232	233	234	235	236	237	238	239	240
30	241	242	243	244	245	246	247	248	249	250	251	252	253	254	255	256	257	258	259	260
31	261	262	263	264	265	266	267	268	269	270	271	272	273	274	275	276	277	278	279	280			C1						B5
32	281	282	283	284	285	286	287	288	289	290	291	292	293	294	295	296	297	298	299	300			R1
33																							T1						G4
34	Important Note: Covered here only snake and diagonal search, for being the more complicated ones.																						T8
35	Anyhow, to cover the rest of them the only thing we have to do is defining new constant arrays																						D12
36	cs (constant snake) {-1,1,20,-20}																						T15
37	cd (constant diagonal) {-21,-19,19,21}
38	For example, for horizontal lookup we have to define																						K8
39	ch {-1,1} and for vertical cv: {-20,20}
40																							O6
41																							M14
42
43
WM snake diagonal

Cell Formulas
Range	Formula
W9,W20,W17,A17,AC12,AC9	W9	=FORMULATEXT(W10)
W10:W15	W10	=WR("snake",cs)
AC10	AC10	=WR("diagonal",cd)
AC13	AC13	=WR("advance",cd)
A18:T32	A18	=SEQUENCE(rw,20)
W18	W18	=WR("difficult",cs)
W21:W22	W21	=WR("winner",cs)
W31:W36	W31	=PR(W10#)
AC31	AC31	=PR(AC10)
AC33,W38	AC33	=PR(AC13)
W40:W41	W40	=PR(W21#)
Dynamic array formulas.

Named Ranges
Name	Refers To	Cells
ar	='WM snake diagonal'!$A$1:$T$15	AC13, AC10, W21, W18, W10

 

[Xlambda]

Few examples before revealing the concept.

Play Free Sudoku online - solve web sudoku puzzles

Play free Sudoku online from Easy to Expert level on Sudoku.com. Select a difficulty level of a web sudoku puzzle to challenge yourself and enjoy the game!

sudoku.com

SDK array.xlsx
	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R	S	T	U	V	W	X
1		extreme sudokus from sudoku.com
2												=SK.2IJ(B3:J11)
3					7	5						5 218	2	8	4	7	5	3	1	9	6
4					2		8		4	5			3	9	6	2	1	8	7	4	5
5			5							2			1	5	7	4	9	6	8	3	2
6			7				1						6	7	2	5	3	1	9	8	4
7		8		1		4	9	2		7			8	3	1	6	4	9	2	5	7
8				9		2		6					5	4	9	8	2	7	6	1	3
9		4	1							8			4	1	5	9	6	2	3	7	8
10				3				4					7	6	3	1	8	5	4	2	9
11				8			4	5	6				9	2	8	3	7	4	5	6	1
12
13												SK.2IJ function solves using recursion and also shows info about recursive loops.
14												2nd value, total loops, 1st value, nr. of iterations of last loop that solved the puzzle.
15
Sheet3

Cell Formulas
Range	Formula
L2	L2	=FORMULATEXT(L3)
L3:U11	L3	=SK.2IJ(B3:J11)
Dynamic array formulas.

SDK array.xlsx
	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R	S	T	U	V
15
16												=SK.2IJ(B17:J25)
17				7		1			6	5		4 76	4	2	7	3	1	9	8	6	5
18		9			7			2					9	1	5	7	8	6	2	4	3
19			8										6	8	3	4	5	2	7	9	1
20			9				1			7			5	9	8	2	6	1	4	3	7
21				4				9	1				7	6	4	8	3	5	9	1	2
22			3										1	3	2	9	7	4	5	8	6
23				1		4			2	9			8	7	1	5	4	3	6	2	9
24						2			5				3	4	9	6	2	7	1	5	8
25			5					3					2	5	6	1	9	8	3	7	4
26
Sheet3

Cell Formulas
Range	Formula
L16	L16	=FORMULATEXT(L17)
L17:U25	L17	=SK.2IJ(B17:J25)
Dynamic array formulas.

SDK array.xlsx
	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R	S	T	U	V
26
27												how many loops for "creating" a solved puzzle from scratch
28												=SK.2IJ(B29:J37)
29												29 403	6	5	3	8	4	9	7	1	2
30													8	9	1	3	2	7	6	4	5
31													2	7	4	1	6	5	3	8	9
32													4	2	8	7	3	6	9	5	1
33													1	6	5	2	9	8	4	7	3
34													9	3	7	5	1	4	2	6	8
35													3	4	9	6	5	1	8	2	7
36													7	1	2	4	8	3	5	9	6
37													5	8	6	9	7	2	1	3	4
38
Sheet3

Cell Formulas
Range	Formula
L28	L28	=FORMULATEXT(L29)
L29:U37	L29	=SK.2IJ(B29:J37)
Dynamic array formulas.

 

[Xlambda]

A simpler solution, for a single function, but this time with 5x5 of 9x9 arrays on which every row and every column of any next 9x9 array has no dups (SUDOKUs have also every next 3x3 arrays with no dups)
It's iterative, 5 in 5 loops of 3M iterations => total 75M iterations
NK(n,k) delivers kxk of nxn arrays Each nxn array has no dups byrow or bycol

=LAMBDA(n, k,
    LET(
        s, SEQUENCE(n),
        q, SEQUENCE(k),
        r, LAMBDA(n, TOROW(SORTBY(s, RANDARRAY(n)))),
        c, LAMBDA(x, AND(BYCOL(x, LAMBDA(y, ROWS(UNIQUE(y)) = ROWS(x))))),
        b, LAMBDA(x,
            REDUCE(
                r(n),
                SEQUENCE(10 ^ 6, 3),
                LAMBDA(v, i, IF(ROWS(v) = x, v, LET(a, VSTACK(v, r(x)), IF(c(a), a, v))))
            )
        ),
        DROP(REDUCE("", q, LAMBDA(v, i, VSTACK(v, DROP(REDUCE("", q, LAMBDA(v, i, HSTACK(v, b(n)))), , 1)))), 1)
    )
)

NNK.xlsx
	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R	S	T	U	V	W	X	Y	Z	AA	AB	AC	AD	AE	AF	AG	AH	AI	AJ	AK	AL	AM	AN	AO	AP	AQ	AR	AS	AT	AU
1
2		=NK(9,5)
3		6	1	8	3	4	5	9	7	2	1	7	9	8	6	3	4	2	5	3	1	2	9	5	4	6	7	8	6	7	2	5	1	8	3	9	4	3	4	6	9	2	5	7	1	8
4		9	2	6	8	5	1	7	4	3	7	9	5	1	4	8	6	3	2	5	4	9	3	7	8	1	2	6	5	6	1	2	7	3	8	4	9	7	5	2	8	4	9	3	6	1
5		4	3	5	7	8	9	6	2	1	5	8	6	4	2	7	3	9	1	7	3	4	8	2	6	9	5	1	7	3	5	6	9	1	4	8	2	1	6	8	5	9	3	2	4	7
6		3	6	1	4	7	8	2	5	9	6	2	7	5	1	9	8	4	3	2	6	7	5	1	9	3	8	4	1	8	9	3	4	6	2	7	5	2	8	9	1	6	7	4	5	3
7		7	5	4	1	9	2	3	6	8	2	3	4	7	9	5	1	8	6	1	2	3	4	9	7	8	6	5	9	2	4	8	3	5	1	6	7	5	1	4	7	8	2	6	3	9
8		2	8	3	9	6	4	5	1	7	9	5	3	6	7	4	2	1	8	4	9	6	7	8	5	2	1	3	3	5	8	4	6	9	7	2	1	6	9	3	2	7	4	1	8	5
9		1	7	9	5	2	3	4	8	6	3	4	1	2	8	6	9	5	7	8	7	5	6	3	1	4	9	2	2	9	6	7	8	4	5	1	3	9	3	7	4	5	1	8	2	6
10		8	4	7	2	3	6	1	9	5	4	1	8	3	5	2	7	6	9	9	8	1	2	6	3	5	4	7	8	4	3	1	2	7	9	5	6	8	2	5	3	1	6	9	7	4
11		5	9	2	6	1	7	8	3	4	8	6	2	9	3	1	5	7	4	6	5	8	1	4	2	7	3	9	4	1	7	9	5	2	6	3	8	4	7	1	6	3	8	5	9	2
12		5	3	9	7	2	4	8	1	6	4	6	2	7	9	3	8	1	5	7	8	6	5	3	4	1	2	9	4	9	5	8	2	1	7	3	6	8	5	7	9	6	1	4	2	3
13		2	1	7	8	3	5	6	4	9	2	1	3	9	5	6	7	8	4	6	5	4	1	9	2	3	8	7	5	2	8	7	6	4	3	1	9	9	3	4	1	5	6	2	7	8
14		7	5	6	1	4	8	9	2	3	6	5	8	1	4	2	9	3	7	1	9	5	8	2	7	4	3	6	2	7	1	5	4	3	6	9	8	5	1	8	2	9	4	6	3	7
15		9	7	1	6	5	3	2	8	4	5	7	4	3	6	8	1	2	9	4	2	7	9	1	5	8	6	3	7	3	6	9	8	2	4	5	1	1	7	9	8	2	3	5	6	4
16		3	9	5	2	8	7	4	6	1	3	8	9	5	1	7	2	4	6	8	3	1	7	5	9	6	4	2	9	5	3	4	1	7	8	6	2	6	2	1	4	3	7	9	8	5
17		8	6	4	3	7	9	1	5	2	1	2	5	6	7	4	3	9	8	9	7	2	3	4	6	5	1	8	6	4	9	2	5	8	1	7	3	3	8	6	7	4	9	1	5	2
18		6	2	8	4	9	1	3	7	5	7	4	1	2	8	9	6	5	3	3	6	8	2	7	1	9	5	4	1	8	2	6	3	5	9	4	7	7	4	2	5	1	8	3	9	6
19		1	4	3	5	6	2	7	9	8	8	9	7	4	3	1	5	6	2	5	4	9	6	8	3	2	7	1	3	6	4	1	7	9	2	8	5	4	6	5	3	7	2	8	1	9
20		4	8	2	9	1	6	5	3	7	9	3	6	8	2	5	4	7	1	2	1	3	4	6	8	7	9	5	8	1	7	3	9	6	5	2	4	2	9	3	6	8	5	7	4	1
21		8	5	7	1	6	3	9	2	4	6	9	7	3	1	4	5	8	2	3	7	1	2	9	4	8	6	5	9	4	5	2	3	8	1	6	7	7	3	5	8	9	1	2	4	6
22		9	3	2	8	4	6	1	5	7	3	4	8	7	9	6	1	2	5	1	8	2	6	7	5	9	3	4	7	6	4	9	1	5	3	2	8	1	7	6	5	3	4	9	2	8
23		2	6	9	7	1	5	4	8	3	2	5	4	8	6	7	9	1	3	2	1	9	4	5	7	3	8	6	5	7	1	6	2	4	9	8	3	6	1	2	9	7	3	4	8	5
24		6	2	4	3	5	9	7	1	8	9	2	1	6	8	5	3	7	4	9	2	4	1	8	6	5	7	3	1	8	3	4	9	6	2	7	5	3	9	8	4	6	2	7	5	1
25		7	9	5	4	3	2	8	6	1	1	7	3	2	4	9	8	5	6	8	9	7	3	6	2	4	5	1	6	2	9	8	4	3	7	5	1	9	6	1	2	4	8	5	3	7
26		1	7	8	2	9	4	6	3	5	8	6	9	1	5	3	2	4	7	7	3	8	5	2	1	6	4	9	4	3	2	7	8	1	5	9	6	2	8	9	7	5	6	3	1	4
27		5	8	1	9	2	7	3	4	6	7	8	2	5	3	1	4	6	9	6	4	5	7	3	8	1	9	2	8	1	7	5	6	9	4	3	2	4	5	3	6	1	7	8	9	2
28		3	4	6	5	8	1	2	7	9	4	1	5	9	7	2	6	3	8	4	5	6	9	1	3	7	2	8	2	9	8	3	5	7	6	1	4	5	2	4	1	8	9	6	7	3
29		4	1	3	6	7	8	5	9	2	5	3	6	4	2	8	7	9	1	5	6	3	8	4	9	2	1	7	3	5	6	1	7	2	8	4	9	8	4	7	3	2	5	1	6	9
30		5	9	2	4	3	6	7	1	8	2	1	4	9	5	8	7	6	3	3	7	4	2	1	5	6	8	9	5	4	6	1	2	8	3	7	9	8	5	6	1	9	2	3	4	7
31		1	3	5	6	4	8	2	7	9	3	9	5	4	8	2	1	7	6	7	3	1	5	9	6	2	4	8	7	2	5	6	8	1	9	3	4	3	9	4	5	8	6	7	2	1
32		6	4	7	9	2	1	8	5	3	8	7	6	5	2	1	3	4	9	5	6	3	1	2	8	9	7	4	3	5	2	7	1	9	8	4	6	6	2	1	4	7	9	5	3	8
33		8	7	6	1	5	9	3	2	4	4	5	3	8	6	9	2	1	7	2	9	7	6	8	4	3	5	1	8	6	1	2	9	4	7	5	3	4	3	5	6	1	8	9	7	2
34		2	1	3	5	8	4	6	9	7	7	8	2	6	1	3	5	9	4	9	4	2	8	6	3	5	1	7	4	7	9	8	3	2	5	6	1	2	8	3	7	5	4	6	1	9
35		7	2	9	8	6	3	1	4	5	5	3	7	2	9	4	6	8	1	1	5	8	3	4	2	7	9	6	2	1	7	4	5	3	6	9	8	5	6	7	9	4	1	2	8	3
36		3	5	4	2	1	7	9	8	6	1	4	8	7	3	6	9	2	5	8	1	5	9	3	7	4	6	2	6	8	3	9	4	7	1	2	5	1	7	8	3	2	5	4	9	6
37		9	8	1	3	7	5	4	6	2	6	2	9	1	7	5	4	3	8	4	8	6	7	5	9	1	2	3	1	9	4	3	6	5	2	8	7	7	1	9	2	6	3	8	5	4
38		4	6	8	7	9	2	5	3	1	9	6	1	3	4	7	8	5	2	6	2	9	4	7	1	8	3	5	9	3	8	5	7	6	4	1	2	9	4	2	8	3	7	1	6	5
39		1	5	9	4	7	2	3	8	6	5	3	2	8	6	9	4	7	1	6	4	7	8	1	2	5	9	3	2	8	6	7	4	5	9	3	1	3	4	7	8	6	5	2	9	1
40		6	1	2	3	4	5	7	9	8	9	2	8	5	1	4	7	3	6	1	8	3	2	9	5	4	6	7	3	7	2	4	6	9	5	1	8	5	1	3	6	8	9	7	2	4
41		2	6	5	7	8	1	9	4	3	4	1	6	2	9	7	3	8	5	4	3	9	5	6	8	2	7	1	9	2	4	5	1	8	6	7	3	7	8	9	5	4	3	6	1	2
42		3	7	1	8	2	4	6	5	9	3	7	9	6	2	8	1	5	4	5	2	8	9	7	1	3	4	6	1	5	3	2	7	4	8	9	6	8	7	4	3	1	2	9	5	6
43		8	9	6	2	5	3	1	7	4	1	6	5	9	7	3	2	4	8	8	9	2	1	3	7	6	5	4	7	4	5	1	8	6	3	2	9	4	6	2	1	5	7	3	8	9
44		7	4	8	1	9	6	2	3	5	8	5	3	7	4	1	6	9	2	7	6	1	4	5	3	9	8	2	4	1	9	6	5	3	2	8	7	2	3	1	4	9	6	5	7	8
45		5	2	3	9	1	8	4	6	7	7	9	4	1	5	6	8	2	3	2	1	4	6	8	9	7	3	5	6	3	8	9	2	7	1	5	4	9	5	8	2	7	1	4	6	3
46		4	3	7	5	6	9	8	1	2	2	4	1	3	8	5	9	6	7	3	5	6	7	2	4	8	1	9	5	9	7	8	3	1	4	6	2	6	2	5	9	3	8	1	4	7
47		9	8	4	6	3	7	5	2	1	6	8	7	4	3	2	5	1	9	9	7	5	3	4	6	1	2	8	8	6	1	3	9	2	7	4	5	1	9	6	7	2	4	8	3	5
48
Sheet2

Cell Formulas
Range	Formula
B2	B2	=FORMULATEXT(B3)
B3:AT47	B3	=NK(9,5)
Dynamic array formulas.

A lambda that has number of iterations "ni" as separate parameter
(I found the solution with real SUDOKU's far more challenging, this is only if someone prefers a sigle cell formula)
NK(n,k,ni) delivers kxk of nxn arrays using a "k" in "k" loops of "ni"x3 iterations for each loop => k*k*ni*3 iterations. (ni grows "mega" exponential with the size of "n" (for n=5, ni=1000 is enough)

=LAMBDA(n, k, ni,
    LET(
        s, SEQUENCE(n),
        q, SEQUENCE(k),
        r, LAMBDA(n, TOROW(SORTBY(s, RANDARRAY(n)))),
        c, LAMBDA(x, AND(BYCOL(x, LAMBDA(y, ROWS(UNIQUE(y)) = ROWS(x))))),
        b, LAMBDA(x,
            REDUCE(
                r(n),
                SEQUENCE(ni, 3),
                LAMBDA(v, i, IF(ROWS(v) = x, v, LET(a, VSTACK(v, r(x)), IF(c(a), a, v))))
            )
        ),
        DROP(
            REDUCE(
                "",
                q,
                LAMBDA(v, i, VSTACK(v, DROP(REDUCE("", q, LAMBDA(v, i, HSTACK(v, b(n)))), , 1)))
            ),
            1
        )
    )
)

Book1
	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R	S	T	U	V	W	X	Y	Z	AA
1
2		=NK(4,3,100)
3		3	4	1	2	4	1	3	2	4	3	2	1
4		2	1	3	4	2	4	1	3	3	4	1	2
5		1	2	4	3	1	3	2	4	1	2	3	4
6		4	3	2	1	3	2	4	1	2	1	4	3
7		4	3	2	1	1	2	3	4	2	3	1	4
8		3	4	1	2	4	1	2	3	1	4	2	3
9		1	2	3	4	2	3	4	1	4	1	3	2
10		2	1	4	3	3	4	1	2	3	2	4	1
11		1	3	2	4	1	4	2	3	2	4	3	1
12		3	2	4	1	4	2	3	1	3	1	2	4
13		4	1	3	2	2	3	1	4	4	3	1	2
14		2	4	1	3	3	1	4	2	1	2	4	3
15
16		=NK(5,5,1000)
17		4	2	3	1	5	5	2	3	4	1	3	2	4	1	5	5	2	3	1	4	2	4	3	5	1
18		2	3	1	5	4	2	3	4	1	5	2	3	1	5	4	1	3	2	4	5	5	1	2	4	3
19		5	1	2	4	3	3	1	2	5	4	5	1	3	4	2	2	4	1	5	3	3	5	1	2	4
20		1	4	5	3	2	4	5	1	3	2	4	5	2	3	1	4	1	5	3	2	1	2	4	3	5
21		3	5	4	2	1	1	4	5	2	3	1	4	5	2	3	3	5	4	2	1	4	3	5	1	2
22		3	2	5	1	4	4	3	1	2	5	1	2	3	4	5	1	3	2	5	4	2	4	3	1	5
23		1	5	3	4	2	5	1	4	3	2	5	4	1	2	3	4	5	1	3	2	1	2	5	3	4
24		2	1	4	5	3	1	2	5	4	3	3	1	4	5	2	3	2	5	4	1	3	1	4	5	2
25		5	4	2	3	1	3	5	2	1	4	2	3	5	1	4	5	1	4	2	3	4	5	1	2	3
26		4	3	1	2	5	2	4	3	5	1	4	5	2	3	1	2	4	3	1	5	5	3	2	4	1
27		2	1	4	5	3	1	4	2	3	5	5	2	3	1	4	2	1	4	3	5	4	3	5	1	2
28		4	2	5	3	1	2	3	1	5	4	2	1	4	3	5	3	4	5	1	2	3	5	1	2	4
29		3	5	2	1	4	4	5	3	2	1	4	3	1	5	2	4	2	3	5	1	1	4	2	3	5
30		1	4	3	2	5	3	1	5	4	2	1	4	5	2	3	1	5	2	4	3	2	1	4	5	3
31		5	3	1	4	2	5	2	4	1	3	3	5	2	4	1	5	3	1	2	4	5	2	3	4	1
32		3	1	2	5	4	5	1	3	4	2	2	1	4	3	5	5	3	4	1	2	3	4	5	1	2
33		1	5	3	4	2	2	3	1	5	4	5	4	3	2	1	3	5	2	4	1	4	1	3	2	5
34		5	2	4	1	3	4	2	5	3	1	1	2	5	4	3	2	1	3	5	4	1	3	2	5	4
35		2	4	5	3	1	1	5	4	2	3	3	5	2	1	4	1	4	5	2	3	5	2	1	4	3
36		4	3	1	2	5	3	4	2	1	5	4	3	1	5	2	4	2	1	3	5	2	5	4	3	1
37		4	5	3	2	1	2	5	4	3	1	2	1	3	4	5	5	1	4	2	3	5	3	1	2	4
38		5	2	4	1	3	5	3	2	1	4	5	4	1	3	2	3	2	1	4	5	4	1	5	3	2
39		1	3	5	4	2	1	4	3	2	5	3	5	2	1	4	1	5	2	3	4	3	2	4	5	1
40		3	1	2	5	4	4	2	1	5	3	1	2	4	5	3	4	3	5	1	2	2	4	3	1	5
41		2	4	1	3	5	3	1	5	4	2	4	3	5	2	1	2	4	3	5	1	1	5	2	4	3
42
Sheet1

Cell Formulas
Range	Formula
B2,B16	B2	=FORMULATEXT(B3)
B3:M14	B3	=NK(4,3,100)
B17:Z41	B17	=NK(5,5,1000)
Dynamic array formulas.

Lambda Functions
Name	Formula
NK	=LAMBDA(n,k,ni,LET(s,SEQUENCE(n),q,SEQUENCE(k),r,LAMBDA(n,TOROW(SORTBY(s,RANDARRAY(n)))),c,LAMBDA(x,AND(BYCOL(x,LAMBDA(y,ROWS(UNIQUE(y))=ROWS(x))))),b,LAMBDA(x,REDUCE(r(n),SEQUENCE(ni,3),LAMBDA(v,i,IF(ROWS(v)=x,v,LET(a,VSTACK(v,r(x)),IF(c(a),a,v)))))),DROP(REDUCE("",q,LAMBDA(v,i,VSTACK(v,DROP(REDUCE("",q,LAMBDA(v,i,HSTACK(v,b(n)))),,1)))),1)))

 

[Xlambda]

NKni.xlsx
	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O
1
2
3		checking: no dups byrow or by col for a single 8x8 array
4
5		=NK(8,1,100000)									=BYROW(B6#,LAMBDA(x,COUNT(UNIQUE(x,1))))
6		7	1	4	8	3	6	2	5		8
7		5	3	6	7	8	4	1	2		8
8		3	4	7	1	6	2	5	8		8
9		8	5	2	3	4	1	7	6		8
10		4	6	1	2	5	7	8	3		8
11		2	7	8	5	1	3	6	4		8
12		6	2	5	4	7	8	3	1		8
13		1	8	3	6	2	5	4	7		8
14
15		=BYCOL(B6#,LAMBDA(x,COUNT(UNIQUE(x))))
16		8	8	8	8	8	8	8	8
17
Sheet2

Cell Formulas
Range	Formula
B5,B15,K5	B5	=FORMULATEXT(B6)
B6:I13	B6	=NK(8,1,100000)
K6:K13	K6	=BYROW(B6#,LAMBDA(x,COUNT(UNIQUE(x,1))))
B16:I16	B16	=BYCOL(B6#,LAMBDA(x,COUNT(UNIQUE(x))))
Dynamic array formulas.

 

[MrExcel]

This is amazing - a collection of valid Sudoku grids!

 

[Xlambda]

This is amazing - a collection of valid Sudoku grids!

Glad you liked it. ✌
Remember we were talking about recursion limitations? Turns out it was some kind of a glitch.
Check this out, made a post about it here:

ARF

ARF !! recursive !! DIY Array Recursive Function kit. Idea triggered by an ExcelIsFun YT video posted yesterday EMT1744 (different topic) . Other functions on minisheet AFLAT. Other functions, never defined before, can be copied directly from minisheet APP and ASU, they follow same syntax as the...

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