Sixty-seven squared
Problem
Evaluate the following:
$67^2$
$667^2$
$6667^2$
$66667^2$
What do you notice?
Can you convince someone what the answer would be to (a million sixes followed by a 7) 2 ?
Getting Started
Student Solutions
Markland from The John Roan School, Gareth; Euen and Alex from Madras College, Scotland; and Chin Siang from Tao Nan School, Singapore; all sent in good solutions .
Jack from The Ridings High School described the pattern:
I noticed that the number of 4s and 8s each increased by 1 for each extra 6 and that the last digit was always a 9. I then predicted that 666667 ² would equal 444444888889 and I was correct. Therefore according to this pattern: (1 million 6s followed by a 7) ² would be written 1000001 4s followed by 1000000 8s followed by a nine.
| 67 2 | = | 4489 |
| 667 2 | = | 444889 |
| 6667 2 | = | 44448889 |
| 66667 2 | = | 4444488889 |
Doing these four calculations by long multiplication shows how this pattern works. If $m$ is the number of sixes in the number that is squared, the pattern is:
( $m$ sixes followed by $7$)$^2 =$ ($(m+1)$ $4$'s followed by $m$ $8$'s followed by a $9$).So
(one million sixes followed by a $7$)$^2 =$ (one million and one $4$'s followed by a million $8$'s followed by a $9$).
| 666 | ... | 666667 | |||
| 666 | ... | 666667 | |||
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|||||
| 4666 | ... | 666669 | (x7) | ||
| 40000 | ... | 000020 | (x60) | ||
| 400000 | ... | 000200 | (x600) | ||
| . | |||||
| . | |||||
| . | |||||
| 40 | ... | 000020 | ... | 000000 | (x 6 x 10 m -1 ) |
| 400 | ... | 000200 | ... | 000000 | (x 6 x 10 m ) |
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|||||
| 444 | ... | 444888 | ... | 888889 | Total |
To prove the result using the sums of series, evaluate
to get
Multiplying out and simplifying this gives
Using
for $p = 2m+1$ and $p = m$, we get
which is written $444\dots888\dots9$ that is as $(m+1)$ $4$'s followed by $m$ $8$'s followed by a 9.