Mathematician’s wife

MATHEMATICIAN Vachaspati Mishra has a beautiful and faithful wife named Bhamati who means a world to him — he has neither seen the world nor his wife. His world is limited to only devising theorems and formulas. His neighbours have never seen him, but the world has heard of him.

"Someone has again created a rangoli on the floor outside my room. Who does it everyday?" Vachaspati says on coming out of his room a morning after spending another sleepless night, writing the penultimate chapter of his book. "These geometric patterns have helped me discover a number of formulas. Tonight, I will find out who makes these," he says.

At night, from his hiding place behind the pillar, Vachaspati watches a woman enter the compound. After making a triangular grid of matchsticks, she is preparing to leave when Vachaspati blocks her way. "Who are you and why do you make these patterns?" he says. "I am your wife and these patterns are to inspire you," she says. "I don’t remember getting married," says the learned man. "It is because you are always absorbed in your work," says the wife. "Prove that you are a mathematician’s wife," says the husband. "How?" says the lady. "The grid that you have made today is beautiful. Can you find out the least number of matches that have to be removed to break all triangles in this size 3 grid of 18 matches?" says the husband.

Find the least number of matches that have to be removed to break all triangles in this size 3 grid of 18 matchsticks

The wife says, "Three matches are needed to make a single triangle. To break it, you can remove any single match." She colours the small triangles in red and blue alternately, so that, no two adjacent triangles are of the same colour. If the grid is of size ‘n’, she now has n*(n+1)/2 triangles in red and n*(n-1)/2 triangles in blue. All upward-pointing triangles are of one colour and all downward-pointing triangles of another colour. No matchstick is part of two small triangles of the same colour, so, at least n*(n+1)/2 matches have to be removed to make all small triangles disappear. "If I do this by removing a match from each of the n*(n+1)/2 triangles of the same colour, so that all matches that are removed point in the same direction, I have removed all matches in that direction, due to which, no triangle can now be formed. For this size 3 grid, 3(4)/2=6 matches must be removed to make all triangles disappear," she says.

Vachaspati says, "Beautiful indeed! I will use this formula to end my book, which reminds me that I still have to dedicate it to someone." He picks up his pen, finishes his book, and writes on page 1 — "Dedicated to my wife". "What is your name dear?" he says to his wife, "Bhamati," replies the woman. On the title page, Vachaspati writes in bold letters — Bhamati.

— Aditya Rishi