# Puzzle Page

## From Dozenal Journal no.8

(Puzzles taken or adapted from Dudeney's books)

The number 3025 (base ten) is written on a piece of paper and this is torn in two: (30), (25).

Note that (30+25)^{2} = 55^{2} = 3025, the original number. Another such number in base ten is 9801.

Here's another, sent in by Valerio Deo (Brazil): (20+25)^{2} = 45^{2} = 2025.

found by solving from the form a^{2} + bx + c = 0.

What about other bases?

Two solutions in base twelve are

*3630

36 + 30 = 66 and 66^{2}= 3630

and

*ET01

ET + 01 = EE and EE^{2} = ET01

## Summary to 11/7/2005

base 3 | base 4 | base 5 | base 6 | base 7 |

22: 2101 | 22: 1210 | 31: 2011 | 23: 1013 | 45: 3114 |

| 33: 3201 | 44: 4301 | 33: 2013 | 66: 6501 |

| | 55: 5401 | |

base 8 | base 9 | base ten | base eleven | base twelve |

34: 1420 | 88: 8701 | 45: 2025 | 96: 8313 | 56: 2630 |

44: 2420 | | 55: 3025 | TT: T901 | 66: 3630 |

51: 3221 | | 99: 9801 | | EE: ET01 |

77: 7601 | | | | |

There's a pattern for the last entry for each base.

For any base, r, (r^{2}-1)^{2} gives r^{4}-2r^{2}+1,

more obvious when written in reverse notation (q.v)

## Anyone volunteer to try three-figure numbers? ....

## Dan's contribution: (1/5/2006)

(NB using ABCDEF for digits, as in hexadecimal)

- base 2: 110001
- base 3: 221001
- base 4: 110100, 332001
- base 5: 234044, 443001
- base 6: 210124, 554001
- base 7: 125151, 151152, 152152, 210151, 615024, 665001
- base 8: 404151, 776001
- base 9: 123210, 346210, 527154, 653110, 887001
- base 10: 494209, 998001
- base 11: 132253, 145259, 282283, 283283, 45A259, 483253, 542235, 705184, 789151, 80A139, AA9001
- base 12: 170294, 4A4294, BBA001
- base 13: 190300, 540300, 741259, A37139, CCB001
- base 14: 1482DB, 69C2DB, DDC001
- base 15: 1DA36A, 3B33B4, 3B43B4, 63036A, B031C9, EED001
- base 16: 14E344, 184369, 3E0400, 420400, 7AA369, 82A344, 9AE2C4, A3C290, D60141, F4C059, FFE001

and he adds:

I've also recomputed the two-digit ones, with slightly different answers.

- base 2: 1001
- base 3: 2101
- base 4: 1210, 3201
- base 5: 2011, 4301
- base 6: 1013, 2013, 5401
- base 7: 3114, 6501
- base 8: 1420, 2420, 7601
- base 9: 5714, 8701
- base 10: 2025, 3025, 9801
- base 11: 1225, 4A25, 6A19, 8313, A901
- base 12: 2630, 3630, BA01
- base 13: 1129, 162C, 1B31, 5031, 592C, 6729, CB01
- base 14: 1832, 3037, 4037, 6232, 8C24, DC01
- base 15: 1331, 7A31, ED01
- base 16: 1C39, 3840, 4840, 7239, A429, FE01

Dan's contributions copied from his post at the DozenalForum