CAT 2024 Slot 3 · Quant · Question 1

The expression \left| x+y \right|+\left| x-y \right| has a neat closed form that is worth knowing.

For any two reals, \left| p \right|+\left| q \right|=\max \left( \left| p+q \right|,\left| p-q \right| \right): when p and q share a sign the larger of the two is \left| p+q \right|, and when they differ in sign it is \left| p-q \right|.

Taking p=x+y and q=x-y, the sum p+q=2x and the difference p-q=2y.

So \left| x+y \right|+\left| x-y \right|=\max \left( \left| 2x \right|,\left| 2y \right| \right)=2\max \left( \left| x \right|,\left| y \right| \right).

The equation therefore says 2\max \left( \left| x \right|,\left| y \right| \right)=2, that is \max \left( \left| x \right|,\left| y \right| \right)=1.

For integers, this means both \left| x \right| and \left| y \right| are at most 1, and at least one of them is exactly 1.

Each of x and y can be -1, 0 or 1, giving 3\times 3=9 pairs in all.

Only \left( 0,0 \right) fails the requirement that one of them equals 1, so it is the single pair to drop.

The count is 9-1=8.

The eight pairs are \left( 1,1 \right), \left( 1,0 \right), \left( 1,-1 \right), \left( 0,1 \right), \left( 0,-1 \right), \left( -1,1 \right), \left( -1,0 \right) and \left( -1,-1 \right).

Shortcut. The identity \left| x+y \right|+\left| x-y \right|=2\max \left( \left| x \right|,\left| y \right| \right) turns the equation into the boundary of a square with corners at \left( \pm 1,\pm 1 \right). Counting lattice points on a square's outline is then instant: all 9 points of the 3\times 3 grid except the centre.