# A number puzzle

I am getting married and quite a few of my guests are mathematicians. I would like to design an invite with a puzzle. Basically, I would like to pose a short puzzle with the answer 26.

My example (but might be too easy (it can be solved with the knowledge of quadratic equation)).

If

$$a+b = 4$$ and $$a^2 + b^2 = 10$$ then $$a^3 + b^3 -2 = ?$$

However, it need not be a number theory question. Thank you for your time.

• Isn't it the only (natural) number that is neighbors with both a square and a cube? May 2, 2017 at 6:06
• Byron starts at the edge of a circular swimming pool and swims 10 meters before hitting the edge. He then turns exactly 90 degrees to the left and swims 24 meters before hitting the edge once again. He is now exactly opposite the center from where he started. How far is he from his starting point? May 2, 2017 at 6:10
• What is the smallest (natural) number that isn't a prime, nor has one within $2$ of itself? May 2, 2017 at 6:13

1) Tell me the only integer that is one greater than a square $(5^2 + 1)$ and one less than a cube $(3^3 − 1)$.
2) Find the smallest number $(26)$ that is not a palindrome whose square is a palindrome $(26^2 = 676)$.