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Nim

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.

Back to Basics

Find b where 3723(base 10) = 123(base b).

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Composite Notions

Age 14 to 16
Challenge Level

The following solution was recieved from Andrei of School 205 Bucharest. Well done and thank you Andrei.
 

10201 could be written (in base $x$) as:

$$\begin{align*}10201 &= 1x^0 + 2x^2 + 1x^4 \\ &= x^4 + 2x^2 + 1 \\ &= (x^2 + 1)^2 \end{align*}$$

Now, I write 10101 in a similar manner, in base $y$:

$$\begin{align*} 10101& = y^4 + y^2 + 1\\ & = y^4 + 2y^2 - y^2 + 1\\ & = (y^4 + 2y^2 + 1) - y^2\\ & = (y^2 + 1)^2 - y^2\\ & = (y^2 + 1 + y)(y^2 + 1 -y) \end{align*}$$

Therefore both expressions can be factorised, so they are composite.