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

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15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

Counting Factors

Age 11 to 14 Challenge Level:

 

Consider the prime factorisation of 12:
$$\begin{align} 12 &= 2 \times 6 \\ &= 2 \times 2 \times 3  \end{align}$$
So $12 = 2^2 \times 3^1$.
 
Can you see how the table below can be used to find the six factors of 12?
 

$2^0$

$3^0$

$3^1$

$2^1$

$3^0$

$3^1$

$2^2$

$3^0$

$3^1$

The first branch gives us $2^0 \times 3^0 =1$ 
The second branch gives us $2^0 \times 3^1 =3$ 
The third branch gives us $2^1 \times 3^0 =2$ 
The fourth branch gives us $2^1 \times 3^1 =6$ 
The fifth branch gives us $2^2 \times 3^0 =4$ 
The sixth branch gives us $2^2 \times 3^1 =12$