Tom, from Wilson's School, found some other
pairs which gave multiples of 1000 and repeated
digits:
Other pairs that give multiples of $1000$ are:
$60^2 - 40^2 = 2000$
$70^2 - 30^2 = 4000$
$75^2 - 25^2 = 5000$
$80^2 - 20^2 = 6000$
$85^2 - 15^2 = 7000$
$90^2 - 10^2 = 8000$
All these pairs add up to 100 which is a factor of 1000.
Other ways to make multiples of $1000$ are $95^2 - 45^2 = 7000$ and
$95^2 - 55^2 = 6000$. These numbers when added together become a
factor of the answer.
Other pairs that give repeated digit answers are:
$67^2 - 34^2 = 3333$
$56^2 - 45^2 = 1111$
The numbers in the tens columns add together to make 9 while the
numbers in the units columns add together to make 11.
Tabitha from The Norwood School, Hannah
from Munich International School, Richard from Comberton Village
College, and Paul, Dulan and Priyan from Wilson's School all
explained how the diagram helped them to work out the difference
between two square numbers. Here is Richard's
explanation:
By drawing a $85 \times 85$ square and a superimposed $65 \times
65$ square, and subtracting the areas, a strip which covers two
lengths of the larger square is formed. When segments of the strip
are rotated, a rectangle is formed that has a constant width of the
difference of the squares' sides and a length of the sum of the
squares' sides.

So the dimensions of the purple rectangle are: width $85 - 65 =
20$, length $85 + 65 = 150$
This implies that the area of the purple strip is equal to that of
the rectangle, $20 \times 150 = 3000$.
In general,by drawing a square of length $x$ with a superimposed
square of length $y$, the area of the strip will be $(x+y)(x-y)$,
so $x^2 - y^2 = (x+y)(x-y)$
James and Nat from Sawston both used this
formula for the difference of two squares to help them to find
pairs of numbers. Hannah used the formula to work out some of the
other things posed in the problem:
$7778^2 - 2223^2 = (7778 - 2223)(7778+2223) = 5555 \times 10001 =
55555555$
$88889^2 - 11112^2 = (88889 - 11112)(88889 + 11112) = 77777 \times
100001 = 7777777777$
Lyman from Nanjing International School
showed how the numbers could be made in lots of different ways by
finding factors, including some examples using decimals. Here are
some ways of making 1000:
$251^2 - 249^2$
$127^2 - 123^2$
$55^2 - 45^2$
$35^2 - 15^2$
$500.5^2 - 499.5^2$
$102.5^2 - 97.5^2$
$66.5^2 - 58.5^2$
$32.5^2 - 7.5^2$
Finally, Richard explained more generally
how to make multiples of 1000 and numbers with repeated
digits:
Any number of the form $n \times 10^3$ where $n$ is an integer, can
be expressed as $x^2-y^2$ when $x + y = 100$ and $x - y =
10n$.
This is not the only case. More generally, it can be formed when $x
+ y = u \times 10^a$ and $x - y = v \times 10^b$, where $u \times v
< 10$ and $a + b = 3$ or where $u \times v$ is equal to some
multiple of $10$ and $a + b = 2$.
This is because $x^2 - y^2 = (x+y)(x-y) = uv \times
10^{(a+b)}$.
A repeated number can also be generated quite easily. A number of
the form $mm$ can be expressed as $x^2 - y^2$ when $x + y = 11$ and
$x - y$ is equal to $m$.
A number of the form $mnmn$ is expressible when $x + y = 101$ and
$x - y = mn$.
$mnomno$ is expressible when $x + y$ is equal to $1001$ and $x - y
= mno$
Well done to all who submitted
solutions.