KenKen
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KenKen (tm) is a mathematical and logical puzzle loosely similar to sudoku. It was invented by a Japanese mathematics teacher Tetsuya Miyamoto, innovator of the theory "The Art Of Teaching Without Teaching", as an 'instruction-free' method of training the brain and was later published in The Times in March 2008. It was introduced to The Times via Robert Fuhrer of Nextoy and Chess Champion Dr. David Levy and recognized for its depth and magnitude by The Times features editor, Mr. Michael Harvey. KenKen(tm) brain training puzzles are a trademark of Nextoy, LLC. Renowned toy inventor, Robert Fuhrer, founder of Nextoy discovered KenKen (aka KEN-KEN) in Japan as the original books published by educational publisher Gakken Co., Ltd. as Kashikoku naru Puzzle, and has been instrumental in introducing them to the western world.
The objective is to fill the grid in with the digits 1 through 6 such that:
- Each row contains exactly one of each digit
- Each column contains exactly one of each digit
- Each bold-outlined group of cells contains digits which achieve the specified result using the specified mathematical operation: addition (+), subtraction (-), multiplication (x), and division (/). Unlike Killer sudoku, digits may repeat within a group.
Some of the techniques from sudoku and killer sudoku can be used here, but much of the process involves the listing of all the possible options and eliminating the options one by one as other information requires. One major distinction of KenKen from sudoku is that there are multiple ways to an answer. One can finish a puzzle and then erase the answers to try to solve it in a more efficient way.
In the example here:
- "11+" in the leftmost column can only be "5,6"
- "2/" in the top row must be one of "1,2", "2,4" or "3,6"
- "20x" in the top row must be "4,5".
- "6x" in the top right must be "1,1,2,3". Therefore the two "1"s must be in separate columns, thus row 1 column 5 is a "1".
- "240x" on the left side is one of "6,5,4,2" or "3,5,4,4". Either way there is a five and it must be in the right pair of cells since we have "5,6" already in column 1.
- etc
More complex KenKen problems are formed using the principles described above but omitting the +-x/ symbol leaving them as yet another unknown to be determined.
[edit] Data Tables
The following table evaluate the various combinations of 2, 3, 4 and 5 cells for each operation. For instance, 4 may be obtained by multiplying the four digits 1, 1, 2 and 2 together.
Where a group of digits includes repeats, then it must be part of a kinked group not a linear one.
1 -12 = 2 *112
1 -23 = 3 *113
1 -34 = 4 +112 4 *114
1 -45 = 4 *122
1 -56 = 5 +113 5 *115
2 *12 2 -13 2 /12 = 5 +122
2 -24 2 /24 = 6 +114 6 *116
2 -35 2 /36 = 6 +123 6 *123
2 -46 = 7 +115
3 +12 3 *13 3 -14 3 /13 = 7 +124
3 -25 3 /26 = 7 +133
3 -36 = 7 +223
4 +13 4 *14 4 -15 4 /14 = 8 +116 8 *124
4 -26 = 8 +125
5 +14 5 *15 5 -16 5 /15 = 8 +134
5 +23 = 8 +224
6 +15 6 *16 6 /16 = 8 +233
6 +24 6 *23 = 9 +126 9 *133
7 +16 = 9 +135
7 +25 = 9 +144
7 +34 = 9 +225
8 +26 8 *24 = 9 +234
8 +35 = 10 +136 10 *125
9 +36 = 10 +145
9 +45 = 10 +226
10 +46 10 *25 = 10 +235
11 +56 = 10 +244
12 *26 = 10 +334
12 *34 = 11 +146
15 *35 = 11 +155
18 *36 = 11 +236
20 *45 = 11 +245
24 *46 = 11 +335
30 *56 = 11 +344
================================================================= 12 +156 12 *126
8 +11222 8 *11222 = 12 +246 12 *134
9 +11223 = 12 +255 12 *223
10 +11224 = 12 +336
10 +11233 = 12 +345
11 +11225 = 13 +166
11 +11234 = 13 +256
11 +11333 = 13 +346
12 +11226 12 *11223 = 13 +355
12 +11235 = 13 +445
12 +11244 = 14 +266
12 +11334 = 14 +356
12 +12333 = 14 +446
13 +11236 = 14 +455
13 +11245 = 15 +366 15 *135
13 +11335 = 15 +456 16 *144
13 +11344 = 16 +466 16 *224
13 +12334 = 16 +556
13 +22333 = 17 +566
14 +11246 = 18 *136
14 +11255 = 18 *233
14 +11336 = 20 *145
14 +11345 = 20 *225
14 +11444 = 24 *146
14 +12335 = 24 *226
14 +12344 = 24 *234
14 +22334 = 25 *155
15 +11256 = 30 *156
15 +11346 = 30 *235
15 +11355 = 32 *244
15 +11445 = 36 *166
15 +12336 = 36 *236
15 +12345 = 36 *334
15 +12444 = 40 *245
15 +22335 = 45 *335
15 +22344 = 48 *246
16 +11266 16 *11224 = 48 *344
16 +11356 = 50 *255
16 +11446 = 54 *336
16 +11455 = 60 *256
16 +12346 = 60 *345
16 +12355 = 72 *266
16 +12445 = 72 *346
16 +13444 = 75 *355
16 +22336 = 80 *445
16 +22345 = 90 *356
16 +22444 = 96 *446
17 +11366 = 100 *455
17 +11456 = 108 *366
17 +11555 = 120 *456
17 +12356 = 144 *466
17 +12446 = 150 *556
17 +12455 = 180 *566
17 +13445 ============================
17 +22346 = 4 *1122
17 +22355 = 6 +1122 6 *1123
17 +22445 = 7 +1123
17 +23444 = 8 +1124 8 *1124
18 +11466 18 *11233 = 8 +1133
= 8 +1223
18 +11556 = 9 +1125 9 *1133
18 +12366 = 9 +1134
18 +12456 = 9 +1233
18 +12555 = 10 +1126 10 *1125
18 +13446 = 10 +1135
18 +13455 = 10 +1144
18 +22356 = 10 +1234
18 +22446 = 10 +2233
18 +22455 = 11 +1136
18 +23445 = 11 +1145
18 +33444 = 11 +1235
19 +11566 = 11 +1244
19 +12466 = 11 +2234
19 +12556 = 12 +1146 12 *1126
19 +13456 = 12 +1155 12 *1134
19 +13555 = 12 +1236
19 +22366 = 12 +1245
19 +22456 = 12 +1344
19 +22555 = 12 +2235
19 +23446 = 12 +2244
19 +23455 = 13 +1156
19 +33445 = 13 +1246
20 +11666 20 *11225 = 13 +1255
20 +12566 = 13 +1345
20 +13466 = 13 +2236
20 +13556 = 13 +2245
20 +14555 = 13 +2344
20 +22466 = 14 +1166
20 +22556 = 14 +1256
20 +23456 = 14 +1346
20 +23555 = 14 +1355
20 +33446 = 14 +2246
20 +33455 = 14 +2255
21 +12666 = 14 +2345
21 +13566 = 14 +3344
21 +14556 = 15 +1266 15 *1135
21 +22566 = 15 +1356
21 +23466 = 15 +1455
21 +23556 = 15 +2256
21 +24555 = 15 +2346
21 +33456 = 15 +2355
21 +33555 = 15 +3345
22 +13666 = 16 +1366 16 *1144
22 +14566 = 16 +1456
22 +22666 = 16 +2266
22 +23566 = 16 +2356
22 +24556 = 16 +2455
22 +33466 = 16 +3346
22 +33556 = 16 +3355
22 +34555 = 17 +1466
23 +14666 = 17 +2366
23 +23666 = 17 +2456
23 +24566 = 17 +3356
23 +33566 = 17 +3455
23 +34556 = 18 +1566 18 *1136
23 +44555 = 18 +2466 18 *1233
24 +15666 24 *11226 = 18 +3366
24 +24666 24 *11234 = 18 +3456
24 +33666 = 18 +4455
24 +34566 = 19 +2566
24 +44556 = 19 +3466
25 +25666 = 19 +4456
25 +34666 = 20 +3566 20 *1145
25 +44566 = 20 +4466
26 +35666 = 21 +4566
26 +44666 = 22 +5566
27 +45666 27 *11333 = 24 *1146
28 +55666 = 24 *1234
30 *11235 = 25 *1155
32 *11244 = 30 *1156
36 *11236 = 30 *1235
36 *11334 = 32 *1244
40 *11245 = 36 *1166
45 *11335 = 36 *1236
48 *11246 = 36 *2233
48 *11344 = 40 *1245
50 *11255 = 48 *1246
54 *11336 = 48 *1344
54 *12333 = 48 *2234
60 *11256 = 50 *1255
60 *11345 = 60 *1256
64 *11444 = 60 *1345
72 *11266 = 60 *2235
72 *11346 = 64 *2244
72 *12334 = 72 *1266
75 *11355 = 72 *1346
80 *11445 = 72 *2236
90 *11356 = 75 *1355
90 *12335 = 80 *2245
96 *11446 = 90 *1356
96 *12344 = 96 *2246
100 *11455 = 96 *2344
108 *11366 = 100 *1455
108 *12336 = 100 *2255
108 *22333 = 108 *1366
120 *11456 = 120 *1456
120 *12345 = 120 *2256
125 *11555 = 120 *2345
128 *12444 = 144 *1466
144 *11466 = 144 *2266
144 *12346 = 144 *2346
144 *22334 = 144 *3344
150 *11556 = 150 *2355
150 *12355 = 180 *1566
160 *12445 = 180 *2356
180 *11566 = 180 *3345
180 *12356 = 200 *2455
180 *22335 = 216 *2366
192 *12446 = 216 *3346
192 *13444 = 225 *3355
192 *22344 = 240 *2456
200 *12455 = 270 *3356
216 *11666 = 288 *2466
216 *12366 = 300 *3455
216 *22336 = 324 *3366
240 *12456 = 360 *2566
240 *13445 = 360 *3456
240 *22345 = 400 *4455
250 *12555 = 432 *3466
256 *22444 = 480 *4456
288 *12466 = 540 *3566
288 *13446 = 576 *4466
288 *22346 = 720 *4566
300 *12556 = 900 *5566
300 *13455
300 *22355
320 *22445
360 *12566
360 *13456
360 *22356
375 *13555
384 *22446
384 *23444
400 *22455
432 *12666
432 *13466
432 *22366
450 *13556
480 *22456
480 *23445
500 *14555
500 *22555
540 *13566
576 *22466
576 *23446
576 *33444
600 *14556
600 *22556
600 *23455
648 *13666
720 *14566
720 *22566
720 *23456
720 *33445
750 *23555
864 *14666
864 *22666
864 *23466
864 *33446
900 *23556
900 *33455
1000 *24555
1080 *15666
1080 *23566
1080 *33456
1125 *33555
1200 *24556
1296 *23666
1296 *33466
1350 *33556
1440 *24566
1500 *34555
1620 *33566
1728 *24666
1800 *34556
1944 *33666
2000 *44555
2160 *25666
2160 *34566
2400 *44556
2592 *34666
2880 *44566
3240 *35666
3456 *44666
4320 *45666
5400 *55666
