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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<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: