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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?


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.

The Genie in the Jar

Age 11 to 14
Challenge Level

Natasha , a student studying at the Indian Language School, in Lagos, Nigeria sent in her solution to another one of our "much-valued problems", The Genie in the Jar. Thank you Natasha for this very comprehensive explanation of your thinking.

1. We find the prime factors of five hundred and ninety-five:
$595 = 5\times 7\times 17$, which are the prime factors.

2. We know that the jar holds enough oil to fill 'granid' silver bottles:
volume of jar = volume of 'granid' bottles

3. If $1$ silver bottle holds enough oil to fill 'ozvik' golden goblets then,
volume of $1$ silver bottle = volume of 'ozvik' golden goblets

Therefore, based on this knowledge, we can make the statement:
volume of 'granid' silver bottles = volume of 'ozvik'$\times$ 'granid' golden goblets.

If we look back to from step 2 we discover that:
volume of the jar = volume of 'ozvik'$\times$ 'granid' golden goblets

4. We can work out that $1$ golden goblet holds enough oil to fill 'vaswik' crystal spoons: volume of $1$ golden goblet = volume of 'vaswik' crystal spoons

So this means that,
the volume of 'ozvik'$\times$ 'granid' golden goblets = volume of 'vaswik'$\times$ 'ozvik'$\times$ 'granid' crystal spoons

Using the information from step 3 we can figure out that,
the volume of the jar = volume of 'vaswik'$\times$ 'ozvik'$\times$ 'granid' crystal spoons.

5. We also know something else important, the oil in the jar lasted for $595$ days.
The oil in vaswik'$\times$'ozvik'$\times$'granid' spoons lasted for $595$ days

If we look back at the information in step 4 we can discover that,
vaswik'$\times$ 'ozvik'$\times$ 'granid' = $595$
vaswik'$\times$ 'ozvik'$\times$ 'granid' = $5\times 7\times 17$

The genie's words: 'granid', ozvik' and 'vaswik' stand for our numbers five, seven and seventeen.

Do you agree with Natasha's thinking and solution? Chris Milliken also worked hard on this problem and had a different strategy from Natasha. Do you see how and why your solution differs too Chris?

Have you solved this problem in a different way? We would like to hear from you.