Crossover

Age 11 to 14

ShortChallenge Level Yellow star

Answer: 7

Using totals

21 + 21 = 42

2 + 3 + 4 + 5 + 6 + 7 + 8 = 35

7 missing so the 7 is couted twice

x = 7

Using totals and symbols

The cells in the diagram can each be assigned a letter. Since the row and column both sum to

$21$ , this gives the equations:

$a+b+x+c = 21$

$d+x+e+f = 21$

Then, the letters

$a$ ,

$b$ ,

$c$ ,

$d$ ,

$e$ ,

$f$ and

$x$ are, in some order,

$2$ ,

$3$ ,

$4$ ,

$5$ ,

$6$ ,

$7$ and

$8$ . This means that their sums must be the same, so;

$a+b+c+d+e+f+x = 2+3+4+5+6+7+8 = 35.$ Then, if the equations at the top are added together, these give:

$a+b+c+d+e+f+2x=42.$

But then, subtracting the previous equation from this one gives

$x = 7$ , as all the other terms cancel out.

This value of

$x$ can be achieved as shown in the diagram on the right.

This problem is taken from the UKMT Mathematical Challenges.

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

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