## Sum of Three Squares

Find the four smallest distinct positive integers $a,b,c,d$ such that $a^2+b^2+c^2=d^2$.
Source: NCTM Mathematics Teacher

SOLUTION
Consider the square integers $1,4,9,16,25,\dots$ We form sets of square integers; find all possible three-element subsets of those sets; and see if the sum of the three elements is a square.
$A=\left \{1,4,9\right \}$
Number of three-element subsets
$\binom{3}{3}=1$
$1+4+9=14$  Not a solution

$A=\left \{1,4,9,16\right \}$
$\binom{4}{3}=4$
$1+4+9=14$  Repeat of previous step
$1+4+16=21$  No
$1+9+16=26$  No
$4+9+16=29$  No

$A=\left \{1,4,9,16,25\right \}$
$\binom{5}{3}=10$
$4$ subsets are repeat; $6$ new ones are listed below
$25+1+4=30$ No
$25+1+9=35$  No
$25+1+16=42$  No
$25+4+9=38$  No
$25+4+16=45$  No
$25+9+16=50$  No

$A=\left \{1,4,9,16,25,36\right \}$
$\binom{6}{3}=20$
$10$ subsets are repeat; $10$ are new
$36+1+4=41$  No
$36+1+9=46$  No
$36+1+16=53$  No
$36+1+25=62$  No
$36+4+9=49$  Yes
$6^2+2^2+3^2=7^2$
Though not necessary we list the remaining $5$ subsets for completeness
$36+4+16=56$  No
$36+4+25=65$  No
$36+9+16=61$  No
$36+9+25=70$  No
$36+16+25=77$  No

Answer: $2,3,6,7$

Retired high school math teacher.
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### One Response to Sum of Three Squares

1. Thomas Egense says:

A similar (but much harder) puzzle is to find 4 square numbers, such that when adding any 3 of them, you also get a square number. A brute force computer search can instantly solve this, I am not sure it can be done by hand.

I wrote a blog post about a similar unsolved problem in recreational mathematics, which I did not solve.
http://www.thomas-egense.dk/math/Squares_on_a_cube.html