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.

What about other bases?

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

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, (r²-1)² gives r⁴-2r²+1,
more obvious when written in reverse notation (q.v)

(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.

Dan's contributions copied from his post at the DozenalForum