Using your Head is Permitted

April 2007 riddle

This is an improvement over a classic riddle. The original riddle goes like this:

Consider the number 3 0 2 5.
It satisfies several interesting properties:

  • It has four digits
  • All four digits are different
  • If we take the number that is the first two digits and sum it with the number that composes the last two digits, its square is the original number: (30+25)2=3025.
The original riddle asked you to find another such number. There is exactly one other, so altogether two. If the requirement that the digits be different is omitted, a third possibility presents itself. Finally, two more solutions enter into the game if we allow the four digits to include leading zeros, so all in all, five solutions to the relaxed version of this riddle.

It is this relaxed riddle that we want to consider. The riddle this month is composed of two questions. Answer both to be considered a solver.

Question 1:

If the relaxed version of the riddle is asked in an arbitrary base, b, instead of in base 10, how many solutions is it going to have, as a function of b?

Question 2:

Consider the strict (original) version of the riddle, but, once again, asked in base b instead of in base 10. Will there be an answer for every b≥4? Prove your claim.

List of solvers:

Oded Margalit (4 Apr 23:00)
Itsik Horovitz (10 Apr 20:46)

Elegant solutions can be submitted to the puzzlemaster at Names of solvers will be posted on this page. Notify if you don't want your name to be mentioned.

The solution will be published at the end of the month.


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