# Problem of the Month (June 2011)

The matrix below has the property that, ignoring spaces, every row and column is a power of 2.

 1 6 6 5 5 3 6 5 1 2 3 2 6 4

What is the smallest symmetric matrix of digits and spaces that contains a row of bn where every row and column is some power of b? (Here "smallest" means fewest entries, or smallest sum of powers for the same sized matrices.) What if the matrices do not need to be symmetric? What if the matrices are not allowed to use any power of b more than once?

We can ask the same question of any sequence of numbers that grows at least exponentially. What about Fibonacci numbers? What about factorials?

Are there non-trivial matrices where every row is a power of b and every column is a power of c? (The trivial examples are 1×1 matrices, when b is a power of c, or "disconnected" matrices that are a combination of smaller results.)

Solvers this month include Joe DeVincentis, Bryce Herdt, Jon Palin, and Berend van der Zwaag.

Joe DeVincentis pointed out that there are many powers that use only two digits, such as 816192 = 6661661161, but here we only consider small bases.

Jon Palin proved that there are solutions for all Fibonacci numbers, factorials, and powers of 2, 3, 4, and 5. Bryce Herdt proved it for all powers of 6. Berend van der Zwaag proved it for powers of 8, 11, 12, 16, and 21. Joe DeVincentis proved it for powers of 51.

Joe DeVincentis conjectured that there are no other solutions for powers of 7 and 9, but was not quite able to prove this.

Here are the best-known solutions for symmetric matrices:

Powers of 2
1
 1
2
 2
4
 4
8
 8
16
 1 1 6
32
 3 2 2
64
 6 4 4
128
 1 2 8 2 8
256
 2 2 5 6 6 4
512
 5 1 2 1 2
1024
 1 1 0 2 4 2 4
2048
 2 2 0 4 8 4 8
4096
 4 1 0 2 4 4 0 9 6 6 4 2 4
8192
 8 1 8 1 9 2 2
16384 (JP)
 1 1 6 1 6 3 8 4 8 4
32768
 3 2 2 3 2 7 6 8 6 4 8 2

65536
 1 6 6 5 5 3 6 5 1 2 3 2 6 4
131072 (BH)
 1 3 2 1 1 0 2 4 1 3 1 0 7 2 2 2 4 2
262144 (BH)
 2 2 6 2 1 4 4 2 1 4 4
524288 (BH)
 5 2 4 2 8 8 2 4 2 8 8
1048576 (JD)
 ``` 1 1 0 24 4 8 25 6 1048576 64 6 4 2 4```
2097152 (JD)
 ``` 8 2 10 24 8 1 9 2 2097152 1 25 6 2 2 4```
4194304 (JD)
 ``` 1 4 1 4194304 4 3 2 1 0 24 4```
8388608 (BZ)
 ``` 8 3 2 8 8 16 8388608 8 2```
16777216 (BZ)
 ``` 3 2 32 12 8 16 16777216 32 7 6 8 32 768 2 1 6 4 8 8 2```

33554432 (BZ)
 ``` 3 2 33554432 51 2 25 6 4 4 32 2```
67108864 (BZ)
 ``` 1 16 67108864 1 1 0 2 4 8 8 6 4 4```
134217728 (BZ)
 ``` 3 2 1 32 4 2 1 134217728 3 2 76 8 2 8 8 2 ```
268435456 (BH)
 ``` 2 1 6 8 4 3 2 268435456 4 25 6 6 4 ```
536870912 (BZ)
 ``` 8 5 1 2 3 2 64 8 536870912 4 09 6 81 9 2 16 2 2 2 ```
1073741824 (BH)
 ``` 3 2 1 10 2 4 1073741824 3 2 3 27 6 8 4 128 8 2 4 4 2```
2147483648 (BZ)
 ``` 2 1 4 2147483648 4 8 3 2 6 4 4 8 ```
4294967296 (BH)
 ``` 4 1 2 8 1 819 2 4 8 1 9 2 1 6 42 94967296 2 81 9 2 6 4 2 2 ```

8589934592 (BZ)
 ``` 8 1 2 8 5 1 2 2 048 409 6 8589934592 32 64 5 12 8 1 9 2 2 2 ```
17179869184 (BZ)
 ``` 1 3 27 68 5 12 3 2 76 8 81 92 12 8 2 56 17179869184 262 14 4 8 6 4 8 8 4 ```
34359738368 (BZ)
 ```1 28 3 5 12 34359738368 4 3 2 2 5 6 8 19 2 327 6 8 3 2 8 3 2 6 4 8 ```
68719476736 (BZ)
 ``` 4 1 6 1 63 8 4 13 1 07 2 1 4 0 9 6 6 4 131 0 7 2 1 4 0 96 6871947 6736 32 6 4 2 6 21 44 4 ```
137438953472 (BZ)
 ``` 4 3 2 1 0 2 4 3 2 137438953472 4 3 2 8 4 0 9 6 25 6 3 2 4 32 7 6 8 2 2 4 ```
274877906944 (BZ)
 ``` 1 32 327 68 1 4 8 3 2 7 6 8 32 7 6 8 27 4877906 944 1 02 4 6 4 6 4 81 9 2 64 6 4 8 8 ```
549755813888 (BH)
 ``` 512 4 819 2 1 549 7 55813888 1 2 5 6 2 5 6 8 1 3 2 8 8 8 ```

Powers of 3
1
 1
3
 3
9
 9
27
 2 7 7 2 9 9
81
 8 1 1
243 (JP)
 7 2 9 2 4 3 3 9
729
 7 2 9 2 7 9
2187
 2 1 8 7 1 8 1 7 2 9 9
6561 (BH)
 1 9 1 1 9 6 8 3 6 5 6 1 1 9 6 8 3 8 1 3 3
19683 (JD)
 1 9 1 9 6 8 3 8 1 3

59049 (JD)
 1 9 7 2 9 6 5 6 1 9 5 9 0 4 9 1 9 6 8 3 2 4 3 9 1 9
177147 (JD)
 1 2 7 2 7 1 2 4 3 1 7 7 1 4 7 3
531441 (BZ)
 ``` 2 7 2 7 531441 3 1 24 3 2 4 3 1 7 29 7 29 9```
1594323 (BH)
 ``` 2 7 1 1594323 9 2 4 3 3 7 2 9 3 ```
4782969 (BZ)
 ```2 4 3 2 7 8 1 7 2 9 9 4782969 9 9 1 3```
14348907 (BZ)
 ``` 1 2 4 3 3 2 4 3 8 1 9 14348907 729 9 1 3 3```

Powers of 4
1
 1
4
 4
16
 1 1 6
64
 6 4 4
256
 1 1 0 2 4 2 5 6 6 4 4
1024
 1 1 0 2 4 2 5 6 4 6 4
4096
 4 1 0 2 4 4 0 9 6 6 4 2 5 6 6 4 4
16384 (JD)
 1 1 6 2 5 6 6 5 5 3 6 1 6 3 8 4 6 4

65536 (JD)
 1 1 6 2 5 6 6 5 5 3 6 1 6 3 8 4 6 4
262144 (JD)
 2 6 2 1 4 4 6 4 2 5 6 1 6 4 4
1048576 (BZ)
 ``` 1 1 02 4 2 5 6 6 4 1 63 8 4 2 5 6 10 48576 25 6 6 4 6 4 4 ```
4194304 (BZ)
 ``` 4 1 4 1 4 0 9 6 4 4194304 1 02 4 6 4 4 ```
16777216 (BZ)
 ``` 1 16 16 16 16777 2 16 16 777 2 16 16 7772 16 2 56 2 56 2 56 16 16 16 6 4```

Powers of 5
1
 1
5
 5
25
 2 5 5
125
 1 1 2 5 5
625
 6 2 5 2 5 5
3125
 3 1 2 5 1 2 5 5
15625
 1 5 1 5 6 2 5 2 5 5
78125 (JD)
 9 7 6 5 6 2 5 7 8 1 2 5 1 6 2 5 5 6 2 5 2 5 5
390625 (BZ)
 ``` 31 25 1 953125 1 39 062 5 15 625 3125 1 2 5 25 5 ```
1953125 (JD)
 1 1 9 5 3 1 2 5 5 3 1 2 5 1 2 5 2 5 5
9765625 (JD)
 9 7 6 5 6 2 5 7 8 1 2 5 1 6 2 5 5 6 2 5 2 5 5

48828125 (JD)
 ```9 765 625 9 7 65 625 9 7 65 625 488 2 8 125 78 1 25 7 8 1 2 5 6 25 5 6 2 5 5 7 81 2 5 6 25 5 125 625 625 625 25 5 ```
244140625 (BZ)
 ``` 31 25 1 95 3 125 2 44 14 06 25 62 5 4 88 2 8 125 4 88 28 125 2 5 2 5 1 2 5 4 882 8125 1 3906 2 5 1562 5 1 3 125 25 5 125 25 25 5 ```
1220703125 (JD)
 ``` 3 12 5 1 9 5312 5 2 5 2 5 1220703 125 78 1 25 39 0 62 5 3125 5 3 125 1 25 12 5 2 5 5 5 5```
6103515625 (BZ)
 ``` 6 2 5 1 6103515625 31 25 5 1 2 5 62 5 25 5 5 ```

Powers of 6
1
 1
6
 6
36
 3 6 6
216
 2 1 6 1 6
1296
 1 2 1 6 1 2 9 6 6 1 6
7776
 7 7 7 6 7 7 7 6 7 7 7 6 6 6 6
46656 (BH)
 ```77 7 6 77 7 6 6 60 466176 6 6 4665 6 6 6 1 77 7 6 6 6 6 ```
279936 (JD)
 2 1 6 2 1 6 2 1 6 2 7 9 9 3 6 1 2 9 6 1 2 9 6 3 6 6 6 6 1 6

1679616 (BH)
 ``` 1 7776 777 6 77 7 6 16 79616 6 1 6 6 6 ```
10077696 (BZ)
 ``` 1 2 1 6 60 46617 6 10077 6 96 777 6 777 6 1 46 6 5 6 6 6 1 16796 16 6 6 ```
60466176 (BH)
 ```77 7 6 77 7 6 6 60 466176 6 6 4665 6 6 6 1 77 7 6 6 6 6 ```
362797056 (BZ)
 ``` 1 3 6 2 1 6 2 1 6 7 7 7 6 1 2 9 6 7 7 7 6 1 00 77696 36279705 6 1 7 77 6 7 7 76 6 1 679616 6 6 6 6 6 ```
2176782336 (BZ)
 ``` 21 6 1 7 77 6 6 7 77 6 2176782336 1 29 6 3 6 36 6 6 6 6 ```

Powers of 7
1
 1
7
 7

Powers of 8
1
 1
8
 8
64 (JD)
 ``` 64 5 12 6 4 4 09 6 549755813888 51 2 5 1 2 8 1 3 2 7 68 8 8 8 1 6 4 8 1 2 6 2 14 4```
512 (JD)
 ``` 64 5 12 6 4 4 09 6 549755813888 51 2 5 1 2 8 1 3 2 7 68 8 8 8 1 6 4 8 1 2 6 2 14 4```
4096 (JD)
 ``` 64 5 12 6 4 4 09 6 549755813888 51 2 5 1 2 8 1 3 2 7 68 8 8 8 1 6 4 8 1 2 6 2 14 4```
32768 (JD)
 ``` 64 5 12 6 4 4 09 6 549755813888 51 2 5 1 2 8 1 3 2 7 68 8 8 8 1 6 4 8 1 2 6 2 14 4```

262144 (JD)
 ``` 64 5 12 6 4 4 09 6 549755813888 51 2 5 1 2 8 1 3 2 7 68 8 8 8 1 6 4 8 1 2 6 2 14 4```
2097152 (JD)
 ``` 51 2 5 1 2 6 4 4 0 9 6 40 9 6 16 7 7 7 216 6 4 2 097 1 5 2 5 12 8 1 5 49 7 55813 888 1 1 2 6 21 4 4 26 2 144 1 6 4 6 4 8 8 8```
16777216 (JD)
 ``` 51 2 5 1 2 6 4 4 0 9 6 40 9 6 16 7 7 7 216 6 4 2 097 1 5 2 5 12 8 1 5 49 7 55813 888 1 1 2 6 21 4 4 26 2 144 1 6 4 6 4 8 8 8```
134217728
?
1073741824
?

8589934592 (JD)
 ``` 5 1 2 6 4 6 4 4 0 9 6 6 4 6 4 6 4 8 5 1 2 8 4 0 9 6 4 0 9 6 5 49 75581 3 888 5 1 2 512 8 1 1 6 4 51 2 4 09 6 858993 4592 2 6 2 1 4 4 1 1 2 6 2 144 6 4 6 4 8 8 8 ```
68719476736 (JD)
 ``` 5 1 2 6 4 6 4 409 6 409 6 409 6 409 6 54 9 755813 8 88 5 1 2 5 1 2 8 1 3 2 7 6 8 1 68 719476736 6 4 1 6 7 7 7 2 1 6 6 4 1 6 7 7 721 6 3 27 6 8 2 6 2 1 44 6 4 2 6 2 14 4 6 4 6 4 8 8 8 8```
549755813888 (JD)
 ``` 64 5 12 6 4 4 09 6 549755813888 51 2 5 1 2 8 1 3 2 7 68 8 8 8 1 6 4 8 1 2 6 2 14 4```

Powers of 9
1
 1
9
 9
81
 8 1 1

Powers of 10
1 (JD)
 1
10 (JD)
 1 1 0
100 (JD)
 1 1 0 1 0 0
1000 (BH)
 1 1 0 1 0 1 0 0 0
10000 (BH)
 1 1 0 1 0 1 0 1 0 0 0 0
Clearly this trend continues.

Powers of 11
1 (JD)
 1
11 (JD)
 1 1 1
121 (JD)
 1 1 2 1 1
1331 (JD)
 1 1 1 3 3 1 1 3 3 1 1 1
14641 (BZ)
 ``` 11 1 21 1 1 14 6 4 1 11 13 3 1 1 61 0 5 1 1948717 1 1 94 8717 1 1 4 6 4 1 214 3588 8 1 1 77 1 56 1 11 1 77 15 6 1 1 4 6 41 1 1 1 1 1 1 1 ```
161051 (BZ)
 ``` 11 1 21 1 1 14 6 4 1 11 13 3 1 1 61 0 5 1 1948717 1 1 94 8717 1 1 4 6 4 1 214 3588 8 1 1 77 1 56 1 11 1 77 15 6 1 1 4 6 41 1 1 1 1 1 1 1 ```
1771561 (BZ)
 ``` 11 1 21 1 1 14 6 4 1 11 13 3 1 1 61 0 5 1 1948717 1 1 94 8717 1 1 4 6 4 1 214 3588 8 1 1 77 1 56 1 11 1 77 15 6 1 1 4 6 41 1 1 1 1 1 1 1 ```
19487171 (BZ)
 ``` 11 1 21 1 1 14 6 4 1 11 13 3 1 1 61 0 5 1 1948717 1 1 94 8717 1 1 4 6 4 1 214 3588 8 1 1 77 1 56 1 11 1 77 15 6 1 1 4 6 41 1 1 1 1 1 1 1 ```
214358881 (BZ)
 ``` 11 1 21 1 1 14 6 4 1 11 13 3 1 1 61 0 5 1 1948717 1 1 94 8717 1 1 4 6 4 1 214 3588 8 1 1 77 1 56 1 11 1 77 15 6 1 1 4 6 41 1 1 1 1 1 1 1 ```

Powers of 12
1 (JP)
 1
12 (JP)
 1 1 2
144 (JP)
 1 1 1 4 4 1 4 4
1728 (BZ)
 ``` 1 1 1 1 2 14 4 1728 1 72 8 1 2 2488 32 1 2 ```
20736 (BZ)
 ``` 1 1 1 1 2 1 2 1 4 4 1 72 8 14 4 1 7 2 8 14 4 1 7 2 8 1 7 28 248 83 2 3583180 8 1 7 2 8 248 8 3 2 1 7 28 248 8 32 20 736 1 2 1 2 2 4 8 8 32 1 2 ```
248832 (BZ)
 ``` 1 1 1 1 2 14 4 1728 1 72 8 1 2 2488 32 1 2 ```
2985984 (BZ)
 ``` 1 1 1 1 1 2 1 2 1 4 4 1 2 1 2 1 4 4 1 2 1 2 1 2 1 4 4 248 8 3 2 2 9 8 5 9 8 4 1 728 2 4 8 8 3 2 1 728 24 8 8 3 2 1 7 2 8 1 2 35831 8 0 8 358 3180 8 29 859 8 4 1 72 8 1 2 2 4 8 8 3 2 1 7 2 8 24 88 32 2 0 736 1 2 1 2 1 4 4 2 4 8 8 32 1 2 ```
35831808 (BZ)
 ``` 1 1 1 1 2 1 2 1 4 4 1 72 8 14 4 1 7 2 8 14 4 1 7 2 8 1 7 28 248 83 2 3583180 8 1 7 2 8 248 8 3 2 1 7 28 248 8 32 20 736 1 2 1 2 2 4 8 8 32 1 2 ```

Powers of 15
1 (JP)
 1
15 (JP)
 1 1 5
225 (JP)
 1 1 2 2 5 2 2 5 1 5 1 5

Powers of 16
1 (JP)
 1
16 (JP)
 1 1 6
256 (JP)
 ``` 1 1 1 1 6 25 6 2 5 6 65536 1 6 1 6 1 6 ```
65536 (JP)
 ``` 1 1 1 1 6 25 6 2 5 6 65536 1 6 1 6 1 6 ```
1048576
?
16777216 (JD)
 ``` 1 1 16 16 16 16777 2 16 16 777 2 16 16 7772 16 2 56 2 56 2 56 16 16 16 1 6 ```

Powers of 21
1 (JD)
 1
21 (JD)
 2 1 1
441 (JD)
 4 4 1 4 4 1 1 1
9261 (JD)
 ``` 1 1 1 944 8 1 44 1 4 4 1 4 08 4 1 0 1 1 4 0 8 4 1 0 1 19448 1 4 4 1 4 4 1 1 9 448 1 1 9448 1 2 1 9261 44 1 441 1 8 010 88 541 4 41 1 1 1 1 1 1 1 1```
194481 (JD)
 ``` 1 1 1 944 8 1 44 1 4 4 1 4 08 4 1 0 1 1 4 0 8 4 1 0 1 19448 1 4 4 1 4 4 1 1 9448 1 1 9 448 1 441 44 1 1 8 010 88 541 4 41 1 1 1 1 1 1 1 1```
4084101 (JD)
 ``` 1 1 1 944 8 1 44 1 4 4 1 4 08 4 1 0 1 1 4 0 8 4 1 0 1 19448 1 4 4 1 4 4 1 1 9448 1 1 9 448 1 441 44 1 1 8 010 88 541 4 41 1 1 1 1 1 1 1 1```

85766121 (JD)
 ``` 1 1 1 9 4481 4 4 1 4 4 1 4 4 1 408 41 0 1 1944 8 1 4 0 8 4 1 0 1 4 0 8 4 1 0 1 1 9448 1 4 41 4 4 1 1 8 01088 5 4 1 1 94 4 8 1 2 1 1 9 4 4 8 1 44 1 2 1 8 5 76612 1 9 26 1 92 6 1 4 4 1 2 1 44 1 4 4 1 1 1 4 0 84 1 01 4 0 8 41 01 4 4 1 1 1 1 1 1 1 1 1 1 1 1```
1801088541 (JD)
 ``` 1 1 1 944 8 1 44 1 4 4 1 4 08 4 1 0 1 1 4 0 8 4 1 0 1 19448 1 4 4 1 4 4 1 1 9448 1 1 9 448 1 441 44 1 1 8 010 88 541 4 41 1 1 1 1 1 1 1 1```

Powers of 38
1 (JD)
 1
1444 (JD)
 ``` 1 1 1 1444 1 444 1 444```

Powers of 51
1 (JD)
 1
51 (JD)
 5 1 1
2601 (JD)
 ``` 13 2 6 51 13 26 51 3 45 02 52 51 5 1 132 65 1 2 6 01 2 60 1 260 1 2 60 1 51 1 3 2 651 6 765 2 01 2601 51 1 3265 1 2601 2 601 51 5 1 2 60 1 51 5 1 51 51 1```
132651 (JD)
 ``` 13 2 6 51 13 26 51 3 45 02 52 51 5 1 132 65 1 2 6 01 2 60 1 260 1 2 60 1 51 1 3 2 651 6 765 2 01 2601 51 1 3265 1 2601 2 601 51 5 1 2 60 1 51 5 1 51 51 1```
6765201 (JD)
 ``` 13 2 6 51 13 26 51 3 45 02 52 51 5 1 132 65 1 2 6 01 2 60 1 260 1 2 60 1 51 1 3 2 651 6 765 2 01 2601 51 1 3265 1 2601 2 601 51 5 1 2 60 1 51 5 1 51 51 1```
345025251 (JD)
 ``` 13 2 6 51 13 26 51 3 45 02 52 51 5 1 132 65 1 2 6 01 2 60 1 260 1 2 60 1 51 1 3 2 651 6 765 2 01 2601 51 1 3265 1 2601 2 601 51 5 1 2 60 1 51 5 1 51 51 1```

Fibonacci Numbers
1
 1
2
 2
3
 3
5
 5
8
 8
13
 1 1 3
21 (BH)
 2 1 1
34
 3 3 4
55
 5 5 5
89
 8 8 9
144
 1 3 4 1 4 4
233
 2 3 2 3 3
377 (BH)
 2 3 3 3 7 7 3 7 7
610 (BH)
 1 6 1 0 8 9 1 4 4 1 0 9 4 6
987
 8 9 8 9 8 7
1597 (BH)
 1 5 8 9 1 5 9 7

2584
 2 5 8 2 5 8 4
4181
 4 1 8 1 1 8 1
6765
 3 6 7 6 5 3 7 7 6 7 6 5 5 5
10946 (BH)
 1 6 1 0 8 9 1 4 4 1 0 9 4 6
17711 (JD)
 1 3 1 7 7 1 1 3 7 7 1 1
28657
 3 2 8 2 8 6 5 7 5 3 7 7
46368
 3 4 4 6 3 6 8 3 1 6 1 0 8
75025 (JP)
 3 3 7 7 5 7 5 0 2 5 2 5
121393 (JP)
 1 2 1 3 1 2 1 3 9 3 3
196418 (JP)
 1 3 8 9 1 9 6 4 1 8 3 4 1 8

317811 (JP)
 3 1 3 1 7 8 1 1 8 1 1
514229 (JP)
 5 1 3 4 2 2 5 1 4 2 2 9
832040 (BH)
 8 3 2 6 1 0 1 4 4 8 3 2 0 4 0
1346269 (BZ)
 ``` 1 3 14 4 1346269 233 46368 9 87```
2178309 (BZ)
 ``` 8 2 1 1 2178309 8 9 3 4 1 0946 8 9 ```
3524578 (JP)
 3 5 2 3 4 5 3 5 2 4 5 7 8 8
5702887 (JP)
 3 5 3 7 7 5 7 0 2 8 8 7 2 8 8 3 7 7
9227465 (BZ)
 ```8 9 233 9227465 37 7 34 67 65 5 5 ```

Factorials
1
 1
2
 2
6
 6
24
 2 2 4
120
 1 2 1 2 0
720
 1 2 7 2 0 2 1 2 0
5040
 1 2 2 1 2 5 0 4 0 1 2 0 2 4 1 2 0
40320
 1 2 1 2 4 1 2 0 4 0 3 2 0 2 4 1 2 0
362880 (BH)
 ``` 1 1 2 2 1 1 2 4 40 3 2 0 40 3 2 0 6 6 2 2 3 6 288 0 3 6 2 880 1 2 0 1 2 0 1 2 0 1 2 0 ```

3628800 (JD)
 ``` 1 2 1 2 1 2 1 1 2 4 40 3 2 0 40 3 2 0 6 6 2 2 3 6 288 0 3 6 2 8800 1 2 0 1 2 0 12 0 12 0 12 0 ```
39916800 (BZ)
 ``` 1 2 1 2 1 2 1 2 1 2 1 1 2 4 40 3 2 0 6 4 0 3 2 0 40 3 2 0 399 1 68 00 3 991 6 800 1 1 6 6 36 288 0 1 2 0 1 2 0 12 0 12 0 12 0 12 0 12 0 ```
479001600 (BZ)
 ``` 1 2 1 2 1 2 1 2 1 2 4 72 0 479001600 1 20 12 0 1 6 12 0 12 0 12 0 ```

Mersenne Primes
3 (JD)
 3
7 (JD)
 7
31 (JD)
 3 3 1
127 (JD)
 3 1 1 2 7 7
131071 (JD)
 ``` 3 3 1 3 31 131071 7 3 1```

Here are the best-known solutions for non-symmetric matrices when they are smaller:

Powers of 2
64
 1 6 4
128
 1 2 8
2048
 1 2 0 4 8 2 4

Powers of 4
64
 1 6 4

Powers of 5
125
 1 2 5 5

Powers of 11
11 (JD)
 1 1

Fibonacci Numbers
13
 1 3
21
 2 1
55
 5 5
144 (BH)
 2 3 3 1 4 4
233
 2 3 3
2584
 3 2 5 8 4
4181
 3 4 1 8 1
17711 (BH)
 3 3 1 7 7 1 1 3 7 7

Here are the best-known solutions for matrices that are not allowed to repeat a row or column:

Powers of 2
1, 4, 16, 64
 1 6 4
2, 8, 128
 1 2 8

Powers of 4
1, 4, 16, 64
 1 6 4

Fibonacci Numbers
1, 3, 13
 1 3
2, 21
 2 1
144 (BH)
 1 3 4 2 5 8 4
233
 1 2 3 3
5, 8, 34, 2584
 3 2 5 8 4
55 (BH)
 3 2 5 8 4 5
4181
 3 2 4 1 8 1

Joe DeVincentis proved that 5 does not have any solutions with even bases.

Here are the best-known solutions for matrices whose rows are powers of b and whose columns are powers of c:

Powers of 2 and 3
(BH)
 8 1

Powers of 2 and 6
(JD)
 1 6
(BH)
 1 2 1 6
(BH)
 1 1 1 2 1 6 4 6 6 5 6 6
(JD)
 ``` 1 1 1 1 1 1 1 6 6 6 777 6 777 6 777 6 216 216 216 6 1 6 ```

Powers of 2 and 9
(BH)
 8 1

Powers of 2 and 11
(JD)
 1 2 1
(JD)
 1 3 3 1 1 2 1 1 2 1
(JD)
 1 1 4 6 4 1
(BH)
 ``` 1 1 21 16105 1 1 2 1 1 46 41```

Powers of 2 and 12
(JD)
 1 2
(JD)
 1 4 4
(BH)
 ``` 1 24883 2 1 358318 0 8 20 736 1 2 1 2 1 4 4 1 2 1 2 1 2 ```
(JD)
 2 4 8 8 3 2 2
 ``` 1 35831808 1 2 1 2 1 2 1 2 1 4 4```

Powers of 2 and 21
(BH)
 `21`
(BH)
 `441`
(BH)
 ``` 1 1 4084101 2 1 2 1 4 4 1```

Powers of 2 and 38
(BH)
 `1444`

Powers of 3 and 8
(BH)
 8 1

Powers of 3 and 11
(JD)
 1 3 3 1
(JD)
 ```1 2 1 1 2 1 1 2 1 19487171 1 3 31 1```
(JD)
 ``` 12 1 1 2 1 1 77 1 561 121 121 121 1 9 4 8 7 1 7 1 1 3 31 16105 1 12 1 1 2 1 1 77 1 5 6 1 121 1 4 6 4 1 121 1 21 1 21 1 9 4 87 1 7 1 1 3 3 1 ```

Powers of 4 and 38
(BH)
 `1444`

Powers of 5 and 15
(JD) (BH)
 2 2 5 1 5 1 5

Powers of 4 and 6
(JD)
 1 6
(BH)
 1 1 1 2 1 6 4 6 6 5 6 6
(JD)
 ``` 1 1 1 1 1 1 1 6 6 6 777 6 777 6 777 6 216 216 216 6 1 6 ```

Powers of 8 and 9
(BH)
 8 1

Powers of 11 and anything
 1 1

Powers of 12 and 38
(JD)
 ``` 1 1 144 1 44 1 44 ```

Powers of 21 and 38
(BH)
 ``` 1 1 44 1 44 1 441```

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 8/6/13.