# Solution to Problem 22

```PROBLEM OF THE WEEK #22
The figure below is a large square with 21 smaller squares contained within. Each of the 21
squares within the large square is a different size with sides of whole number length. Determine
a value for the side length of all 21 inside squares. Yes, n is that little tiny square sandwiched in
between f, g, m, and o.
n
Solution proposed by Philippe Fondanaiche
On the basis of the above picture, we can express 18 sides of squares depending on the three
dimensions of the squares a, n and o and we get the following identities:
b = 20a – 94n – 111o
c = – 2a + 11n + 15o
d = 22a – 105n – 126o
e = – 24a + 116n + 141o
f = 5a – 23n – 27o
g = 5a – 22n – 27o
h = – 46a + 221n + 267o
i = 51a – 243n – 294o
j = 32a – 151n – 182o
k = 6a – 27n – 31o
l = 2n + 3o
m=n+o
p = – 19a + 92n + 112o
q = n + 2o
r = 13a – 59n – 70o
s = 6a – 29n – 34o
t = – 6a +32n + 39o
u = 12a – 56n – 65o
With a computer program written in Basic language, we test the small values of n and o in the
interval 1≤n&lt;o≤10 and the higher values of a 20≤ a≤60 such that all the 18 variables are positive
and distinct.
Finally there exists an unique triple (a,n,o) providing positive and distinct values of the 18
variables with a = 50, n = 2 and o = 7 and we have the well-known solution found by A.J.W.
Duijvestijn in 1978:
```