Squared Triangle

$100+64+16$ .

Proof that there is only this solution.

Only possible squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169.

If ( $a^2+b^2=c^2$ ) then ( $c=s^2+t^2$ ) Also a square can only be of the form $4n$ or $4n+1$

Each square will leave a remainder from 180 which must be the sum of two squares.

Therefore we cannot have odd numbers

1 : odd

4 : 176 is 16*11, which has no solutions ( $4a^2+4b^2=4n$ so 11 would have to be a sum of two sqrs)

9 : odd

16 : 164

25 : odd

36 : 144 is square. But 12 cannot be expressed as sum of two squares (since 3 can't)

49 : odd

64 : 116

81 : odd

100 : 80

121 : odd

144 : 36 is square. But 6 has no solutions

169 : odd

So it could only be 16, 64 and 100. Which do add up to 180

Using 3 perfect squares of 100, 64, and 16, which is $10^2, 8^2,$ and $4^2$ , they will add up to 180.

Starting from 169 (no triangle can have a > 180 degree angle), you can use simple proof by exhaustion in order to prove that 100 + 64 + 16 is the only possibility.

169 - No perfect squares add up to 11 (9+1 is the closest possible) 144 - No perfect squares add up to 36, as it is a perfect square itself 121 - No perfect squares add up to 59 (9 + 49 is closest possible) 100 - 64 + 16 works.

Therefore the solution is 100.

Meet In The Middle approach :v

Sum of angles of triangle is 180 degrees a^2+b^2+c^2 = 180 You know a,b,c < 14 because 14^2 = 196

Testing 13^2 ---> 180-13^2 = 11 and no two squares combine to make 11 Testing 12^2 ---> 180-12^2 = 36 and no two squares combine to make 36 Testing 11^2 ---> 180-11^2 = 59 and no two squares combine to make 59 Testing 10^2 ---> 180-10^2 = 80

a^2+b^2 = 80

80 = 4^2+8^2

Therefore, the largest angle of this triangle is 10^2 degrees = 100 degrees

My method: 180 can write into square number times constant in 2 way, they are 9*20 and 4 \times 45 .

I check out only square 1 4 9 16 25 36 .

for 9*20 I can't write 20 with sum of three square numbers.

but in case of 4*45.

I can make 45 by 4+16+25.

so, 180 = 4 45 = 4 4 +4 16 +4 25 =4^2+8^2+10^2.

therefore 10^2 = 100 is the largest angle.