Business Irish

Friday 22 March 2019

Dominoes fell into place and made a mathematician of me

Cathy O’Neil’s enthusiasm for maths was fuelled by working out puzzles with dominoes on a chess board.
Cathy O’Neil’s enthusiasm for maths was fuelled by working out puzzles with dominoes on a chess board.

Cathy O'Neil

When I was about nine years old, my father taught me how mathematical proofs work. At a time when governments around the world are struggling to narrow the gender gap in maths and science, I figure the experience is worth sharing - because I've never felt more empowered by anything since.

Both my parents are mathematicians. As they raised me and saw what a maths nerd I was becoming, they did a good job of not forcing it.

I even remember the moment I decided to become a mathematician. My dad and I were playing with dominoes on a chess board. The tiles were just the right size to cover two squares, one white and one black.

First he asked me if we could cover the whole chessboard.

Of course, I said. The board had eight rows of eight squares each, so I could just cover each row with four tiles laid end to end - effectively solving eight independent problems. My dad found that convincing.

But what if, he asked, we cut the board down to seven rows of seven squares? I thought for a moment. You couldn't just fit tiles along each row independently. You could put down three, but there'd be a square left over. You could transfer that extra square to the next row, but that would work only if there were an even number of rows. With seven rows, you'd have an extra square sticking out.

In other words, the tiles demonstrated visually why odd numbers aren't divisible by even ones. The seven-by-seven board had 49 squares. Since 49 is odd, I said, you cannot tile such a board with dominoes. Each domino takes up two squares, so they can tile only boards with even numbers of squares.

Satisfied with this answer, he asked me one last question. What if we remove opposite corners of the original eight-by-eight chessboard?

OK, I thought. Take away two squares and you have 62.

That's an even number, so it passed the first test, I figured out. But on the board, I couldn't get it to work. Cutting the board down to four by four didn't help - I still had either uncovered squares or tiles sticking out.

Then I noticed something in the pattern of the board.

Because the squares alternated between black and white in every direction, the tiles always covered one of each. This meant the whole board could be tiled only if it had equal numbers of white and black squares.

But when my dad removed the opposite corners of the chessboard, he'd taken away two white squares.

"You can't tile this," I proclaimed, "because there are more black squares than white squares. Nobody can tile this."

"How much do you want to bet?" said my dad. "Do you want to bet two weeks' allowance that I can't tile this right now? If you win, I'll give you $20." My allowance at the time was $3 per week... I chickened out.

"You should have made the bet, Cathy," he said. "What you just described was a proof. And when you prove something you should put your money behind it. You just lost 20 bucks!"

I was hooked. As the maths I worked on became increasingly complicated and difficult, I doubted my proofs plenty, but I never doubted the beauty and power of logical argument. And I never again lost a bet to my dad. (Bloomberg)

Irish Independent

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