• # Thread: Triangle Puzzle

1. ## Triangle Puzzle

 Arrange the numbers 1 to 9 as shown in the diagram so that the square of the sum of numbers on each of the three sides are equal. Any good ways to do this or does it rely solely on trial and error?

2.  It was a lot simpler after I thought about it for a second. You're trying to find out what pattern would work so that they are all even, right? Well first you have to know the number that you're trying to achieve in order for you to be able to solve it. Just add up all the numbers then do some trial and error.

3.  Do you mean the sum of the squared numbers? If so, why would you even start out with those numbers, and not just throw the final numbers on there right away. I'm confused. I would probably put 9 (81) on one of the corners, and then work from there with small numbers on the other two corners. Caretaker's idea might be good, although I'm not sure if it accounts for the fact that some of the sides share numbers, due to the corners. I'm not sure if that needs to be accounted for either though.

4.  So in the example, the bottom edge has value 1^2 + 2^2 + 3^2 + 4^2, for instance? In this case as Marvo says, the first step is to just square all of the numbers. I'm not sure why he puts 81 on a corner though, this isn't necessarily the solution (in fact, as far as I can tell, it definitely isn't a solution)..? The best hint I can give is to consider the remainders of each square after division by small numbers; for instance, after division by 2. Try to use the fact that the numbers and their sums are either even or odd. Edit: Looks like you've changed the puzzle? Previously it was quite interesting and there was only one answer. Spoiler for Answer: I did it by considering the puzzle modulo 2, 3, 4, and 5. It's then a matter of routine to do various tricks to narrow down what it has to be. The answer is (2, 4, 9, 5), (5, 1, 6, 8), (8, 3, 7, 2). As it stands now the squaring is totally pointless, and I'm wondering where you got the puzzle from. If the squares of the sides are equal then obviously you just have to make it so that the sides are equal, disregarding the squaring.

5.  I can't think of a good way to solve this problem. What I mean is that If for some reason I cared about the actual solution, I would rather write a program to just brute force it than spend a quarter of the time winnowing down possibilities. In the version where the problem incorporates the square of sums rather than the sum of squares (i.e. for the broken version), it can be stated as a non-homogeneous system of 3 linear equations in 9 variables with the RHS equal to (x, x, x) and addressed systematically. I'll try to come up with a good (sorry Xei) solution and report back if I get one.

6.  What's wrong with my solution? :/ I didn't spend any time winnowing it down, my method determined the answer uniquely. How would you work through your solution for the broken system? And why isn't x a variable...

7.  Originally Posted by Xei I'm not sure why he puts 81 on a corner though, this isn't necessarily the solution (in fact, as far as I can tell, it definitely isn't a solution)..? I wasn't sure if I got the puzzle right, so I didn't want to put too much time into thinking about it. I thought putting 81 at a corner would be better, because if you put it on a side, that side would be considerably larger than the other sides very quickly.

8.  Originally Posted by Xei What's wrong with my solution? :/ I didn't spend any time winnowing it down, my method determined the answer uniquely. Except there's not a unique answer to be determined. The answers come in multiples of 48... How would you work through your solution for the broken system? And why isn't x a variable... I wouldn't because my solution wouldn't work. X was intended to be a variable though.

#### Posting Permissions

• You may not post new threads
• You may not post replies
• You may not post attachments
• You may not edit your posts
•