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Adding All Nine

Age 11 to 14 Challenge Level:

Make a set of numbers that use all the digits from $1$ to $9$, once and once only.

1, 2, 3, 4, 5, 6, 7, 8, 9.

For instance, we could choose:

$638, 92, 571$ and $4$

Add them up:

$638 + 92 + 571 + 4 = 1305$

$1305$ is divisible by $9$ (it is $145\times 9$)

(use a calculator to check this if you do not know yet how to divide by $9$)

Add each of the digits in the number $1305$ . What is their sum?

Or, perhaps we could choose:

$921, 4357$ and $68$

Add them up:

$921 + 4357 + 68 = 5346$

$5346$ is divisible by $9$ (it is $594 \times 9$)

Now try some other possibilities for yourself!

I wonder what happens if we use all $10$ digits from $0$ to $9$, once and once only?

0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Try some for yourself!

What do you think would happen if we used the eight digits from $1$ to $8$?

Test your hypothesis by trying some possibilities for yourself!

Were you correct?

Is there a pattern beginning to emerge? Do you have theory that might explain what is happening?

Try some different sets of digits for yourself!