Re: SMSU Problem Corner (oops!)

From: mjc (mjcohen_at_acm.org)
Date: 09/23/04 

Date: Thu, 23 Sep 2004 10:56:16 -0700

Les Reid wrote:
> The original Challenge problem posted had a number of grievous errors. It
> has been withdrawn and a new, similar (and one hopes, correct) puzzle has
> been posted at
>
> http://math.smsu.edu/~les/Challenge.html
>
> Sorry about that!
>
Unless I made a mistake (p>.5), there is no solution to the problem.

The only 4-digit squares with the same first and last digits
and 5-digits triangular numbers with the same first and last digits
both having the same second-highest digit seem to be as shown here:

VEIR is a perfect square
SECHS is triangular
NEUN is a perfect square
What is DREI?
Squares from 1000 through 9999
Look from 32 through 99
68 squares
Tris from 10000 through 99999
Look from 141 through 446
306 triangular numbers
Squares with same first and last digits
39 -> 1521
41 -> 1681
68 -> 4624
75 -> 5625
97 -> 9409
Tris with same first and last digits
141 -> 10011
146 -> 10731
153 -> 11781
158 -> 12561
161 -> 13041
166 -> 13861
173 -> 15051
178 -> 15931
181 -> 16471
186 -> 17391
193 -> 18721
198 -> 19701
257 -> 33153
262 -> 34453
277 -> 38503
282 -> 39903
325 -> 52975
329 -> 54285
330 -> 54615
334 -> 55945
345 -> 59685
348 -> 60726
351 -> 61776
356 -> 63546
363 -> 66066
368 -> 67896
371 -> 69006
407 -> 83028
412 -> 85078
Possible # 1: square 39->1521, tri 412->85078
Possible2 # 1: square 39->1521, tri 412->85078 , square 39->1521
Possible2 # 2: square 39->1521, tri 412->85078 , square 50->2500
Possible2 # 3: square 39->1521, tri 412->85078 , square 81->6561
Possible2 # 4: square 39->1521, tri 412->85078 , square 87->7569
Possible # 2: square 75->5625, tri 181->16471
Possible2 # 5: square 75->5625, tri 181->16471 , square 40->1600
Possible2 # 6: square 75->5625, tri 181->16471 , square 41->1681
Possible2 # 7: square 75->5625, tri 181->16471 , square 51->2601
Possible2 # 8: square 75->5625, tri 181->16471 , square 60->3600
Possible2 # 9: square 75->5625, tri 181->16471 , square 68->4624
Possible2 # 10: square 75->5625, tri 181->16471 , square 75->5625
Possible2 # 11: square 75->5625, tri 181->16471 , square 93->8649
Possible2 # 12: square 75->5625, tri 181->16471 , square 98->9604
Possible # 3: square 97->9409, tri 329->54285
Possible2 # 13: square 97->9409, tri 329->54285 , square 38->1444
Possible2 # 14: square 97->9409, tri 329->54285 , square 49->2401
Possible2 # 15: square 97->9409, tri 329->54285 , square 59->3481
Possible2 # 16: square 97->9409, tri 329->54285 , square 67->4489
Possible2 # 17: square 97->9409, tri 329->54285 , square 74->5476
Possible2 # 18: square 97->9409, tri 329->54285 , square 80->6400
Possible2 # 19: square 97->9409, tri 329->54285 , square 92->8464
Possible2 # 20: square 97->9409, tri 329->54285 , square 97->9409
Possible # 4: square 97->9409, tri 330->54615
Possible2 # 21: square 97->9409, tri 330->54615 , square 38->1444
Possible2 # 22: square 97->9409, tri 330->54615 , square 49->2401
Possible2 # 23: square 97->9409, tri 330->54615 , square 59->3481
Possible2 # 24: square 97->9409, tri 330->54615 , square 67->4489
Possible2 # 25: square 97->9409, tri 330->54615 , square 74->5476
Possible2 # 26: square 97->9409, tri 330->54615 , square 80->6400
Possible2 # 27: square 97->9409, tri 330->54615 , square 92->8464
Possible2 # 28: square 97->9409, tri 330->54615 , square 97->9409

None of the second squares with the second-highest digit matching
have digits distinct from previously occurring ones.

Martin Cohen

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