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An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore. ### Have You Got It?

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

Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers?

# Missing Digits

##### Age 11 to 14 Short Challenge Level:

Working it out by doing the calculation
Starting as normal for multiplication, 4$\times$4 = 16, so P must be 6. Both Ps are replaced by 6s in the calculation below, and the 1 that is carried (from the 16) is also shown. Now, 6$\times$4 + 1 = 25, so N must be 5. Both Ns are replaced by 5s in the calculation below, and the 2 that is carried (from the 25) is also shown. Now, 5$\times$4 + 2 = 22, so M must be 2. Both Ms are replaced by 2s in the calculation below, and the 2 that is carried is also shown. Now, 2$\times$4 + 2 = 10, so L must be 0. Both Ls are replaced by 0s in the calculation below, and the 1 that is carried is also shown. Now, 0$\times$4 + 1 = 1, so K must be 1.

So K, L, M, N, P are 1, 0, 2, 5, 6.

Using algebra
Suppose we are looking for the number KLMNP, and let KLMNP $=x$.
Then the top number, KLMNP4, is KLMNP0 + 4, which is $10x+4$.
The bottom number, 4KLMNP, is 400 000 + KLMNP, which is $400000+x$.

So $(10x+4)\times4=400000+x$. Solving for $x$,\begin{align}40x+16&=400000+x\\39x+16&=400000\\39x&=399984\\x&=399984\div39=10256\end{align}
So K, L, M, N, P are 1, 0, 2, 5, 6.

You can find more short problems, arranged by curriculum topic, in our short problems collection.