# 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 = 552 = 3025, the original number. Another such number in base ten is 9801.

Here's another, sent in by Valerio Deo (Brazil): (20+25)2 = 452 = 2025.
found by solving from the form a2 + bx + c = 0.

Two solutions in base twelve are
*3630
36 + 30 = 66 and 662= 3630
and
*ET01
ET + 01 = EE and EE2 = ET01

## Summary to 11/7/2005

base 3base 4base 5base 6base 7
22: 210122: 121031: 201123: 101345: 3114
33: 320144: 430133: 201366: 6501
55: 5401
base 8base 9base tenbase elevenbase twelve
34: 142088: 870145: 202596: 831356: 2630
44: 242055: 3025TT: T90166: 3630
51: 322199: 9801EE: ET01
77: 7601

There's a pattern for the last entry for each base.
For any base, r, (r2-1)2 gives r4-2r2+1,
more obvious when written in reverse notation (q.v)

## 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

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