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Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}. ### Novemberish

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

# Two and Two

### Why do this problem?

This problem offers an opportunity to practise addition in a more interesting and challenging context than is usual. It requires students to work systematically, record their progress efficiently and apply their understanding of place value.

### Possible approach

This printable worksheet may be useful: Two and Two.

Write the following sum on the board and ask students to complete it.

 8 6 1 + ? ? ? 1 4 2 7

Expect justifications for any suggestions.
This should be unproblematic so move onto a more challenging problem.

If each letter stands for a distinct digit what are the values of $a$, $b$ and $c$?

 b b + c b a c c

"How can we approach this?"
Expect fairly random suggestions to start with but aim to use the discussion as an opportunity to model working systematically.
"What happens if $b$ is $1, 2, 3, \ldots$ ?" - rejecting values as soon as it is apparent they do not work and discussing how you know.
"How can we be sure we have all the solutions?"

Set the students off in pairs to work on the problem TWO + TWO = FOUR.
Establish that you are not going to announce how many solutions there are and that you will expect students to work systematically and be able to justify that they have all the possible solutions.

To finish off, students could present their approach to the rest of the class, with emphasis on explaining clearly why they are convinced that they have found all the possible solutions.

### Key questions

What must "F" be?
What does that tell you about "T"?
Are you certain you have considered all the possibilities?

### Possible extension

Suggest students find other word sums that work.

Here are some possibilities that they might consider:

ONE + ONE = TWO
ONE + TWO = THREE
ONE + THREE = FOUR
FOUR + FIVE = NINE

Why are some impossible?

Can they make a word subtraction?

Students could also try Cryptarithms.