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Playing with Blocks

Abstract. Children's blocks are used to derive some useful mathematical formulas. Do the results hint at Fermat's simple proof?

Fig. 1 shows 52 blocks arranged in a square.

Figure 1

52 Blocks

If we remove 42 blocks, we obtain Fig. 2

Figure 2

52  42 Blocks

 

Fig. 3 shows Fig. 2 with 32 blocks added.

Figure 3

52  42 + 32 Blocks

 

Fig. 4 shows Fig. 3 with 22 blocks removed.

Figure 4

52  42 + 32 22 Blocks

 

Finally, Fig. 5 shows Fig. 4 with 12 blocks added.

Figure 5

52  42 + 32 22 + 12 Blocks

 

Counting the blocks in the horizontal and vertical rows of Fig. 5, it is clear that

5 + 4 + 3 + 2 + 1 = 52  42 + 32 22 + 12. (1)

Starting with 42 blocks, it can similarly be shown that

4 + 3 + 2 + 1 = 42 - 32 +22 - 12. (2)

In general, if N is odd then

N + (N 1) + (N 2) + + 1 = N2  (N 1)2  + 1. (3)

And if N is even then

N + (N 1) + (N 2) + + 1 = N2  (N 1)2  - 1. (4)

Note that, as N grows without bound, Fig. 5 approaches a right triangle of area N2 / 2. Whence we conclude that in the limit of arbitrarily large N,

N2  (N 1)2  + 1 = N2 / 2. (5)

Or, equivalently, in the limit,

N + (N 1) + + 1 = N2 / 2. (6)

It is readily demonstrated that, for certain integer values of N and M, with N > M, N2 - M2 produces integer squares. For example, in Fig. 2, 52  42 results in 9 = 32 blocks. More generally, certain integer values of N and a, with N > a and M = N - a, produce perfect squares for N2  (N - a)2. Other combinations of N and a do not produce such a perfect square result.

It would also seem that exponents greater than 2 cannot produce any "perfect" results at all a proposition first suggested by Fermat. For example, if M = N - a, then

N3  M3 = 3aN(N - a) - a3. (7)

According to Fermat, there are no integers a and N such that the right side of Eq. 7 has an integer cube root. Similar remarks presumably apply to higher integer exponents. In general Fermat claimed to have a simple proof that there is no integer I, such that

Nn  Mn = In, n an integer >2 and N, M integers. (8)

His proof was either never written down, or if so was never found. Countless doodlers have tried since his death to figure out what he might have had in mind. Perhaps he remembered playing with blocks when he was a smart little kid!