Which perfect square of a 4-digit number has the same last four digits as the original number?

Continue reading “Four Digits Squared”# Number Theory

Math puzzles and brain teasers involving the properties of integers

## Arrange the Digits

There are 362,880 different ways to arrange the digits 1 through 9 into a 9-digit number. How many of these combinations are prime numbers?

Continue reading “Arrange the Digits”## Deducing Bicycle Spokes

A bicycle shop has some number of unusual bicycles. All their bikes are identical (there’s more than one bike), and each bike’s front and back wheels has at least one spoke each (each bike has more than one spoke). You don’t know how many bicycles are in the shop, but you know there are between 200 and 300 spokes in total.

If you knew the exact number of spokes, you would be able to figure out the number of bicycles.

You don’t know the exact number of spokes, but just knowing the fact that you would be able to figure it out with that information allows you to deduce the answer. How many bicycles and spokes are there?

Continue reading “Deducing Bicycle Spokes”## Divisible by 100

Prove that for any set of one hundred whole numbers, either one of those numbers is divisible by 100 or the sum of a subset of those numbers is divisible by 100.

Continue reading “Divisible by 100”## Seven Times the Sum of Its Digits

Which two-digit numbers are exactly equal to seven times the sum of its digits?

Continue reading “Seven Times the Sum of Its Digits”## Flipping Every N-th

There are 100 face-down cards. The first person that passes by flips over every card, the 2nd person that passes by flips over every 2nd card, and so forth – the n-th person that passes by flips over every n-th card. Before you know it, 100 people have passed by.

After all 100 people have passed by, which cards are face-up and which are face-down?

Fun fact, this was an actual brainteaser given to me in the first round interview for a hedge fund internship back in 2011.

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Continue reading “Flipping Every N-th”