Problem #1: What is the smallest square that starts and ends with the same n digits?
For example, 16262² = 2644 5 2644 starts and ends with the same 4 digits.

Problem #2: What is the smallest square that contains c copies of the same block of n digits?
For example, 250125² = 625 625 15 625 contains 3 copies of the block 625.

Problem #3: What squares contain cn digits, and when considered as c blocks of n digits, contain only 2 different blocks?
For example, 727727² = 529 586 586 529 contains only the blocks 529 and 586.

Problem #4: What are the answers to these questions in other bases?

Problem #5: What are the answers to these questions for higher powers?

Problem #1:

Joseph DeVincentis, Philippe Fondanaiche, Patrick De Geest, and Richard Sabey were interested in when a square could be 2 identical blocks of n digits. This can clearly only occur when 10ⁿ+1 has a square factor. This happens for n=11, 21, 33, 39, ....

Richard Sabey sent the most detailed analysis: Pick any prime p, with certain exceptions. Find the period 2n of the decimal (or, generalizing, base-b) expansion of 1/p², thus proving that p² divides b²ⁿ–1 = (bⁿ+1)(bⁿ–1). If the period of the expansion of 1/p doesn't divide n, that is, p doesn't divide bⁿ–1, then p² divides bⁿ+1. Then N = a(bⁿ+1)/p where a is chosen so as to make N²/(bⁿ+1) an n-digit number. From this, the primitive solution using prime p and base b, more can be found by replacing n by any odd multiple of n.

Patrick De Geest noted that 36363636364² = 13223140496 13223140496 and 63636363637² = 40495867769 40495867769 and 36363636364 + 63636363637 = 100000000001.

Here are the smallest squares that begin and end with the same n digits. These were all found by Luke Pebody. Evgeni Lukin and Patrick De Geest also sent some solutions.

Problem #2:

Philippe Fondanaiche sent some solutions. Here are the smallest squares with c equal blocks of length n:

Problem #3:

Here are the smallest squares with only m blocks of length n, 2 of which are unique.

n / m	4	5	6	7	8	9	10
1	1 4 4 4 7 7 4 4	1 1 8 8 1 2 9 9 2 9 4 4 9 4 4 5 5 2 2 5 6 9 6 9 6	?	9 6 9 6 9 9 6	?	?	6 6 6 1 6 6 1 1 6 1
2	25 25 06 25 45 45 45 64 80 80 21 21 82 82 82 01 97 97 04 04	16 16 36 16 16 29 29 29 91 29 36 36 81 36 36
3	100 100 100 996 104 104 313 104 141 001 001 001 165 165 836 836 231 625 625 625 352 352 649 649 456 456 979 456 529 586 586 529 564 004 004 004 997 997 004 004	100 100 225 100 100
4	6100 8299 8299 6100 7876 3626 3626 7876 8121 4361 4361 8121 (Philippe Fondanaiche)	1296 1296 2916 1296 1296 (Patrick De Geest)
5	40496 21487 21487 40496 (Philippe Fondanaiche)
6	453289 359866 359866 453289 (Philippe Fondanaiche)

Problem #4:

Scott Reynolds found the smallest solutions for other bases. His data can be found here.

Problem #5:

Richard Sabey found that 7³ = 111 and 14³ = 888 in base 18. These appear to be the only rep-digit powers in other bases.

Here are the smallest cubes that begin and end with the same n digits.

Here are the smallest cubes with c equal blocks of length n:

Another problem concerning patterns in squares can be found here.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 10/31/04.