Can you even the odds? The Wknd Puzzle by Dilip D’Souza

Subtract, replace, repeat. In this week’s numbers game, integers going all the way from 1 to 2022 keep you guessing.

Your mathematically inclined friend Abhilasha hands you a large sheet of paper. On it, she has written the numbers 1, 2, 3, 4..., every integer, all the way to 2022.

“What’s this,” you ask. “Why’d you do it?”

“I want to have a bet with you,” she says, handing you an eraser and putting away her pencil. “Pick any two numbers at a time, erase them and replace them with their difference. Keep doing this.”

“So, at each step, the count of numbers decreases by one,” you ask, feeling clever indeed.

“Yes!” says Abhilasha.

“OK, what’s the bet?”

“That you will finish with a single zero on the page.”

Question: Should you take Abhilasha’s bet?

Hint: At any given step, if you pick one even and one odd number, the replacement will be odd. If both are even or both odd, the replacement will be even.

Another way to think about this: if the pair’s sum is odd, the replacement is odd. If their sum is even, the replacement is even. In mathematical terms, this operation keeps the “parity” of the sum of the pair “invariant”: even to even, odd to odd. What does this tell us?

Scroll down for the answer.

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Answer:

0 is even. Thus the sum of the pair that produces the final 0 must be even; the sums of the pairs that produced each number of that pair must be even, and so on. Work backwards like this and you find that the sum of all the original numbers must be even. Only then will you end with a zero.

But the sum of the numbers from 1 to 2022 is (2022 x 2023) / 2 = 1011 x 2023. Whatever that is, we know it ends in 3, and thus is odd. Therefore you cannot end with a single zero. Therefore you should definitely take Abhilasha’s bet. Happy winning!