Playing with Blocks

G.R.Dixon, 11/11/06

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.