Find as many numbers as you can that have the pattern shown here:
52 – 5 = 42 + 4
72 – 7 = 62 + 6
72 – 7 = 62 + 6
Pretty easy, eh? Write a formula to see why.
There are so many solutions that it’s impossible not to find one, or to find one that doesn’t work.
Another sample: 102 – 10 = 92 + 9.
This can be analyzed in two ways:
102 – 10 = 92 + 9
100 – 10 = 81 + 9
100 – 10 = 81 + 9
OR
102 – 10 = 92 + 9
10(10 – 1) = 9(9 + 1)
10 · 9 = 9 · 10
10(10 – 1) = 9(9 + 1)
10 · 9 = 9 · 10
At the formula level, we have
n2 – n ≟ (n – 1)2 + (n – 1)
= n2 – 2n + 1 + n – 1 = n2 – n, o.k.
= n2 – 2n + 1 + n – 1 = n2 – n, o.k.
OR
n2 – n ≟ (n – 1)2 + (n – 1)
n(n – 1) ≟ (n – 1)((n – 1) + 1) = (n – 1)n, o.k.
n(n – 1) ≟ (n – 1)((n – 1) + 1) = (n – 1)n, o.k.