Question

The instructions below apply to this question and the previous one.

In the figure below, the seven letters correspond to seven unique digits chosen from 0 to 9. The relation among the digits is such that:

\(P \times Q \times R = X \times Y \times Z = Q \times A \times Y\)

P		X
Q	A	Y
R		Z

The sum of the digits which are not used is:

• A. 8
• B. 10
• C. 14
• D. None of the above

Hint:

Once the seven used digits are pinned down from the previous question's working, list which digits from 0 to 9 are left over and add them up.

Answer 1

The Correct Option is D

Add up every digit from 0 to 9 first: $0+1+2+3+4+5+6+7+8+9 = 45$.

The seven digits actually used in the puzzle, found from solving the product condition $P \times Q \times R = X \times Y \times Z = Q \times A \times Y$, are $1, 2, 3, 4, 6, 8, 9$, with $A=2$, $Q=9$, $Y=4$, $\{P,R\}=\{1,8\}$, $\{X,Z\}=\{3,6\}$, common product $72$.

Add up just these seven used digits: $1+2+3+4+6+8+9 = 33$.

Subtracting the used digits' sum from the total of all ten digits gives the sum of the unused ones: $45 - 33 = 12$.

Checking the options: 8, 10 and 14 are all too far off, and 12 is not 15 either, so none of the first four choices equal 12.

So the correct answer is None of the above.