CAT 2025 Slot 2 · Quant · Question 7

The expression measures the total squared gap between a and the other three, so it is least when b, c and d sit as close to a as possible.

All four equal would give 0, but that needs 4a=46, so a=11.5, not an integer.

A value of 1 would need the four numbers to be a,a,a,a\pm 1, summing to 4a\pm 1=46, so 4a=45 or 47 — neither is divisible by 4.

A value of 2 needs a,a,a\pm 1,a\pm 1, summing to 4a\pm 2=46, so 4a=44 and a=11.

Taking a=11, b=11, c=12, d=12, the sum is 46 and the expression is {0}^{2}+{\left( -1 \right)}^{2}+{\left( -1 \right)}^{2}=2.

Since 0 and 1 are impossible and 2 is attained, the minimum is 2.

Shortcut. 46\div 4=11.5, so the four integers must split as two 11s and two 12s to stay as tight as possible. Choose a to be one of the repeated values, and only the two numbers on the other side of 11.5 differ from it — each by 1 — giving 1+1=2.