SMSU Problem Corner (oops!)

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 It has been withdrawn and a new, similar (and one hopes, correct) puzzle has been posted at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

I fear that you misspelled VIER.

I found a solution and, unless I overlooked something, it's unique.

No. Here's why:

1) Don't top post. If the thing needs context, put the context first. If it doesn't need context, snip. 2) You entirely fail to understand what

means.

3) Congratulations for spoiling the puzzle for anyone else who reads sci.math after you.

HAND,