Parity

Two siblings, Martin and Maria, were born exactly one year apart and are celebrating their birthday today. Maria says the sum of their ages is 112. Is that possible?

Born a year apart, their ages are two CONSECUTIVE numbers. Try 51 and 52: that sums to 103. Try other consecutive pairs and something becomes clear.

The counting numbers 1, 2, 3, 4, 5, … alternate between even and odd. So in any two consecutive numbers, one is always even and the other always odd.

What does an even number plus an odd number give? The even part arranges into pairs; the odd part leaves one counter over; so the whole thing has one counter left over. Their sum is ODD.

112 is even. Since Martin's and Maria's ages are consecutive, their sum must be odd. So the ages cannot add to 112, and Maria has made a mistake.

We use the word PARITY for the property of being even or odd. Every whole number has a parity: it is either even or odd, never both. So: the parity of the sum of any two consecutive numbers is odd; the parity of the sum of any two odd numbers is even.

The idea answers many questions at once. Lakpa has an odd number of ₹1 coins, an odd number of ₹5 coins and an even number of ₹10 coins, and counts ₹205 — odd × odd value plus odd × odd value gives an even total, and the ₹10 coins add an even amount, so his total must be EVEN, and 205 is not. He has made a mistake. It works for grids too: a 27 × 13 grid has an odd number of small squares because odd × odd is odd, while 42 × 78 has an even number — no multiplication needed.