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Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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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!

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Have You Got It?

Can you explain the strategy for winning this game with any target?


Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Tim from Gravesend Grammar School and Mohammad Afzaal Butt both sent us similar solutions to the problem. Well done Tim and Mohammad. Here is Mohammad's solution:

Let the three digit number be $xyz$. Hence the six digit number will be $xyzxyz$. Now
$$\eqalign { xyzxyz  &= 100000x + 10000y + 1000z + 100x + 10y + z \cr
&= 100100x + 10010y + 1001z \cr
&= 1001 (100x + 10y + z) \cr
&= 7 \times 11 \times 13 (100x + 10y + z)} $$ Hence the number $xyzxyz$ is always divisible by $7$, $11$ and $13$.