Some Square Pegs and a Round Hole
Murray, Jenny, Mathematics Teaching
Jenny Murray documents an autobiographical ramble through some interesting numbers
A certain experience has given me a lifelong interest in squares and related numbers. Long ago, when I was at school, I "discovered" for myself about successive squares and odd numbers. In great excitement I hunted out my maths teacher, Miss Kemp, to tell her. She told me an equation, which I promptly forgot only to recreate it years later for myself. I now have a suspicion that she had prepared for this in a previous lesson because it came up in class fairly soon afterwards. When it did, I put up my hand and asked,
"What about the difference between cubes?"
"That", she answered firmly, "I will leave for you to find out for yourself, Jenny."
It was before the advent of calculators and I did not get very far! See later, "The round hole" A.
The differences between squares can be seen easily by those who prefer pictures, in this drawing which many children have done with little understanding.
Another question I asked myself and others was, "What number can a square end in?" That is... what will the units be?
What I found out was very interesting. with units
1 and 9 the squares end in 1
2 and 8 the squares end in 4
3 and 7 the squares end in 9
4 and 6 the squares end in 6
5 the squares end in 5
Why, do these pairs add to ten?
For another example of pairs that make ten, see later, "The round hole" B.
That 122 is 144 we have all known for years but to my amazement I found out that 212 is 441 , and 1 32 is 1 69, and 312 961 . These neat reversals seem to go on no longer, but I suspect this may be because of our counting system, rather than anything else. But really an idea that will lead well into ... The round hole!
The round hole is where certain ideas, not all about squares, have fallen in.
A. The differences between successive cubes landed here!
When I had a small child to look after I began to get bored, and decided to explore this some more. ... Still no calculator.
I found that the difference between 1 and 8
(2^sup 3^ minus 1^sup 3^) is 7, between 8 and 27
(3^sup 3^ minus 2^sup 3^) is 19, between 27 and 64
(4^sub 3^ minus 3^sup 3^) is 37, between 64 and 125
(5^sup 3^ minus 4^sup 3^) is 61.
My powers of multiplication and subtraction ran out at this point!
7,19,37,61 ....All prime numbers!
A cousin who was a mathematician at the Open University came to see us. I showed him my short list of primes. He had a calculator and found the next number,
"91 is not a prime," he said dismissively. See later D.
I later did the equation (a + 1 )3 - a3 = 3a2 + 3a + 1 .
Like Miss Kemp, long ago, I will leave the reader to check this! …