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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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Is there an efficient way to work out how many factors a large number has?

Stair Climb

Age 11 to 14 Short
Challenge Level

Answer: Spike, Boris, Perceval


Scaling the number of stairs run
Boris:     5 steps                  100 - finished
Spike:    4 steps       $\times$20      80 - finished (21 + 80 $\gt$ 99)
Percival: 3 steps                   60 - not finished (38 + 60 $\lt$ 99) Last place

                              $\times$19      95 (4 steps to go)
                                           76   (21 + 76 = 97 $\Rightarrow$ 2 steps to go) winner


Working out how long each person takes
Boris:    $99$ steps
Spike:    $99 - 21 = 78$ steps
Percival: $99 - 38 =61$ steps

In each time unit, Boris runs $5$ steps, Spike runs $4$ steps, Percival runs $3$ steps.

Boris:    $99$ steps, $99 \div 5 = 19\frac{4}{5}$
Spike:    $78$ steps, $78 \div 4 = 19\frac{1}{2}$
Percival: $61$ steps, $61 \div 3 = 20\frac{1}{3}$

Therefore they finish in the order Spike, Boris, Percival.

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.