Crossover
The numbers 2 to 8 are to be placed in the diagram so that the row and column each adds to 21. Which numbers can replace x?
Age
11 to 14
Challenge level
Problem
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The numbers $2$, $3$, $4$, $5$, $6$, $7$ and $8$ are placed in the squares in this diagram.
The four numbers in the horizontal row add up to $21$, and the four numbers in the vertical column also add up to $21$.
Which number should replace $x$?
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Student Solutions
Answer: 7
Using totals
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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
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$a+b+x+c = 21$
$d+x+e+f = 21$
$d+x+e+f = 21$
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:$$a+b+c+d+e+f+2x=42.$$
Image
This value of $x$ can be achieved as shown in the diagram on the right.