__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 5^{2} blocks
arranged in a square.

Figure 1

5^{2} Blocks

If we remove 4^{2} blocks,
we obtain Fig. 2

Figure 2

5^{2} –
4^{2} Blocks

Fig. 3 shows Fig. 2 with 3^{2} blocks
added.

Figure 3

5^{2} –
4^{2} + 3^{2} Blocks

Fig. 4 shows Fig. 3 with 2^{2} blocks
removed.

Figure 4

5^{2} –
4^{2} + 3^{2} –2^{2} Blocks

Finally, Fig. 5 shows Fig. 4 with 1^{2} blocks
added.

Figure 5

5^{2} –
4^{2} + 3^{2} –2^{2} +
1^{2} Blocks

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

5 + 4 + 3 + 2 + 1 = 5^{2} –
4^{2} + 3^{2} –2^{2} +
1^{2}. (1)

Starting with 4^{2} blocks,
it can similarly be shown that

4 + 3 + 2 + 1 = 4^{2} -
3^{2} +2^{2} -
1^{2}. (2)

In general, if N is odd then

N + (N – 1) + (N – 2) + … + 1 = N^{2} –
(N – 1)^{2} … + 1.
(3)

And if N is even then

N + (N – 1) + (N – 2) + … + 1 = N^{2} –
(N – 1)^{2} … - 1.
(4)

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

N^{2} –
(N – 1)^{2} … __+__ 1
= N^{2} / 2. (5)

Or, equivalently, in the limit,

N + (N – 1) + … + 1 = N^{2} /
2. (6)

It
is readily demonstrated that, for certain integer values of N and M, with N >
M, N^{2} - M^{2} produces
integer squares. For example, in Fig. 2, 5^{2} –
4^{2} results in 9 = 3^{2} blocks.
More generally, certain integer values of N and a,
with N > a and
M = N - a,
produce perfect squares for N^{2} –
(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

N^{3} –
M^{3} = 3aN(N
- a)
- a^{3}.
(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

N^{n} –
M^{n} = I^{n}, 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!