# Missing number puzzle: how to get this answer?

Today I went to pick up my daughter from school before the start of the winter holidays. I found her chatting with her teacher and they presented me with a little math puzzle. It was on a card from a set of cards with little math puzzles that the teacher had bought somewhere. Most other puzzles were simple, sensible math puzzles, and they had solved them all, except this one. And it left me stumped too.

The puzzle was simple: find the missing number:

3 4 11
5 16 29
6 25 ??


The design on the card had a strong triangle theme, with the numbers in each triplet being arranged in a triangular pattern, and all three triplets were printed on the vertices on a triangle as well. However that could also be just an aesthetic choice.

I didn't inspect the pack too closely, but the puzzles seemed to be meant for children around 10-12 years of age, around grades 4-6. So nothing too fancy - no trigonometry, geometry, complex numbers or whatever. Really simple, basic stuff, with just a smidgeon more of logic and trickiness added, since they were meant to be "hard" problems (for the age group).

The pack also had an answer card, and the correct number in the spot is supposed to be

41

However it did not include any explanations why so. Just an answer. I've been thinking about this all day long, and I cannot figure it out. Best I've come up with is:

a + b + 2*sqrt(b) = c

However that seems way too convoluted and probably is just a coincidence. I've also noted that

The second number is always a perfect square, and the last number is a prime

But whether that has any significance - probably not.

So - any ideas? :)

• Note also that b = (a-1)^2, and c = a(a+1)-1. b seems extraneous, but you could probably shoehorn a lot of different patterns in here. Dec 21, 2019 at 0:49
• @Skynet_0 - Yes, that's the thing I usually don't like about those "missing number" puzzles. With enough pareidolia, there's no end to different answers. Dec 21, 2019 at 1:20

Mathematically: $$a^2 + \sqrt{b}=c$$