IPMAT Indore 2024 · Quantitative Ability · Question 6

Step 1 — shift to the origin. Put u=x-5 and v=y-5. Since x,y are integers, so are u,v, and the condition becomes

\left| u \right|+\left| v \right|\le 6

Shifting does not change how many lattice points there are.

Step 2 — count column by column. Fix u with \left| u \right|\le 6. Then \left| v \right|\le 6-\left| u \right|, which allows

2\left( 6-\left| u \right| \right)+1\text{values of}v

Step 3 — add up.

\sum_{u=-6}^{6}{\left[ 2\left( 6-\left| u \right| \right)+1 \right]}=13+2\sum_{u=1}^{6}{\left[ 2\left( 6-u \right)+1 \right]}

=13+2\,\left( 11+9+7+5+3+1 \right)=13+2\left( 36 \right)=85

Answer: 85

Shortcut. Learn the diamond formula. The number of lattice points with \left| u \right|+\left| v \right|\le n is

2{n}^{2}+2n+1

For n=6: 2\left( 36 \right)+12+1=85. (Sanity check at n=1: the formula gives 5 — the origin plus its four neighbours. ✔) The -5 shifts are pure decoration.