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Adding All Nine

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!

Double Digit

Choose two digits and arrange them to make two double-digit numbers. Now add your double-digit numbers. Now add your single digit numbers. Divide your double-digit answer by your single-digit answer. Try lots of examples. What happens? Can you explain it?

Repeaters

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

Two Many

Age 11 to 14
Challenge Level

What is the least square number which commences with six two's? If you can't at first see how to do this with six twos try three twos. Clearly 222 is not a perfect square. There is no whole number between the square root of 2220 and the square root of 2229 but 149 lies between the square root of 22200 and the square root of 22299.

This was solved by Ling Xiang Ning, Allan, age 13 from Raffles Institution, Singapore and Tony Cardell and John Lesieutre, age 14 from State College Area High School, Pennsylvania, USA.

Here is Allan's solution: First I try 222222. It is not a square number.

Next, I try 2222220 and 2222229. I take the square root of both of them and find out that there is no square number between them.

Next, I try 22222200 and 22222299. I take the square root of both of them and find out that there is no square number between them...

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I try 222222000000 and 222222999999. I take the square root of both of them and find out that there is one square number between them. This number is the square of 471405 which is 222222674025.

So the least square number starting with six twos is 222222674025