GeekDad Puzzle of the Week Solution – Regular Palindrome Numbers

Reading Time: 2 minutes

This past week’s puzzle, as previously posted:

Regular numbers are numbers that evenly divide powers of 60. As an example, 602 = 3600 = 48 × 75, so both 48 and 75 are divisors of a power of 60. They are reasonably rare — under 108, there are just over 1100 of them.

Symmetric or “palindrome” numbers are numbers that read forward and backwards as the same number: in bases 10 and 2, the decimal number 33 (aka 1000012) reads the same in both directions.
mirror_number
This week’s GeekDad Puzzle of the Week is straightforward: How many decimal numbers under 108 are regular, and are also palindromes in at least one base from 2 to 10?

NOTE: Leading zeros don’t count towards symmetry, so 0123032103 is not symmetric in base 3.

There are a couple of different definitions of “regular numbers,” and in the above I gave one of the more obtuse. If you noticed that 60 breaks down into a pair of 2s, a 3, and a 5, you would see that an alternate definition for “regular numbers” are numbers that also are made up of only 2s, 3s, and 5s. The examples of 48 and 75 are 24x3 and 3×52.

Given this definition, it would be straightforward to identify the 9891 or so numbers under 108 that meet this definition. Looping through each one in the 9 different bases from 2..10 to see which values were also palindromes in that base should have been similarly straightforward!

Congratulations to Matt Kelly for coming up with the correct answer of 79 values. His response from drawn at random the the large number of correct responses sent in this past week! A $50 Gift Certificate from the team over at ThinkGeek will be on its way to him shortly!

Matt also noticed that a large number of these regular palindromes were palindromes in base 7. I smell another puzzle coming on around these!

The 79 regular numbers that were also palindromes (and their respective bases) appear below. For numbers that are less than 10, for example, they are palindromes in any base higher than their value, as well as the base just one lower than their value (i.e., 4 is “4” in bases 5-10 as well as “11” in base 3.) The highest multi-base regular palindrome was 2000, which is 55557 and 26629.

1 = 12, 13, 14, 15, 16, 17, 18, 19, 110
2 = 23, 24, 25, 26, 27, 28, 29, 210
3 = 112, 34, 35, 36, 37, 38, 39, 310
4 = 113, 45, 46, 47, 48, 49, 410
5 = 1012, 114, 56, 57, 58, 59, 510
6 = 115, 67, 68, 69, 610
8 = 223, 117, 89, 810
9 = 10012, 118, 910
10 = 1013, 224, 119
12 = 225
15 = 11112, 334
16 = 1213, 227
18 = 335, 228
20 = 2023, 229
24 = 445, 337
25 = 1214
27 = 110112, 338
30 = 339
32 = 447
36 = 1215, 448
40 = 11113, 557, 449
45 = 1011012, 558
48 = 667
50 = 1017, 559
54 = 668
60 = 669
64 = 1217
72 = 2425
80 = 22223, 2126, 889
81 = 1218
100 = 102013, 2027, 1219
125 = 13314
128 = 2427
135 = 3436, 2527
150 = 21124, 3037
160 = 122213, 4246
162 = 2428
192 = 3637
200 = 4047, 2429
216 = 13315
243 = 3638
250 = 5057
300 = 6067, 4548, 3639
400 = 1122113, 11117, 4849
512 = 13317
729 = 13318
750 = 232324
800 = 22227
1000 = 13319
1024 = 26627
1200 = 33337
1458 = 26628
1600 = 44447
1944 = 54457
2000 = 55557, 26629
2400 = 66667
2500 = 102017
3200 = 122217
4096 = 146417
5000 = 204027
6400 = 244427
6561 = 146418
6750 = 254527
7500 = 306037
9600 = 366637
10000 = 146419
20000 = 1122117
25600 = 1344317
40000 = 2244227
60000 = 3366337
125000 = 10303017
160000 = 12343217
250000 = 20606027
1000000 = 113333117
2000000 = 226666227
3515625 = 153223518
6250000 = 1040604017
8000000 = 1246664217
50000000 = 11446644117

Thanks again to everyone that posted a puzzle.

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