## Resources

Here you will find interesting puzzles and problems to keep your kids and the whole family thinking. These problems are a sample of what we do in math circle. For more information or to submit a solution, email us at info@cambridgemathcircle.org. We will run a raffle every 2 weeks with all the correct submissions, and one lucky winner will get a board game!

For those looking for more math, we are holding online classes.

Click on the buttons to or

## A family affair

In one family, each brother has as many sisters as brothers. Each sister in this family has twice as many brothers as she has sisters. How many boys and how many girls are there in this family?

SOLUTION

We can try different small numbers of sisters and brothers. One sister and two brothers is not enough, because the girl doesn't have any sisters. Two sisters and three brothers don't work, because while each boy has an equal number of sisters and brothers, each girl doesn't have twice as many brothers as sisters. Finally, three sisters and four brothers do the trick!

## A prime deal

Three prime numbers (numbers that are only divisible by 1 and themselves) add to 14. To get to the biggest number from the middle number, you need to add the smallest number. What are the numbers? (Hint: the first few primes are 2, 3, 5, 7, 11, and 13.)

(More challenging version) 3 prime numbers add up to 100. One of them is more than a third but less than a half of another number. Find the numbers.

SOLUTION

Since we are adding three primes and getting an even number, one of the primes has to be even. The only even prime is 2, so the 3 numbers must be 2, 5 and 7.

Answer: 2, 5 and 7.

(More challenging version) Again, one of the primes has to be 2, since the sum is even. The other two primes must add to 98. We know that primes ending in 3 won't work, because then the other prime would have to end in 5. Looking at the list of primes below 100, 31 and 67 satisfy the conditions.

Answer: 2, 31, and 67.

## A treasure chest

A treasure chest in a dark cave contains 20 green emeralds and 9 red rubies. What is the smallest number of stones you need to take out, in order to definitely have 2 rubies?

SOLUTION

In the worst case, we will take out every single emerald before starting on the rubies, so it will take 20 + 2 = 22 stones.

Answer: 22 stones.

## A building problem

Tim the Beaver is cutting a long log into shorter logs. He first makes 7 cuts in the long log. He then makes 7 cuts in each of the smaller logs he gets. How many short logs does he end up with?

SOLUTION

We might think that the answer is 7 times 7 or 49, but that is not correct. In fact, it is 8 times 8, since 7 cuts make 8 smaller pieces. Draw it on a piece of paper to convince yourself.

Answer: 64 short logs.

## A climbing problem

Beyonce takes 5 minutes to walk up from the 1st floor to the 5th. How long will she take to go from the first floor to the 9th? She's doing it while singing at the top of her lungs, and that's why it takes so long.

SOLUTION

It may seem that 9 is the right answer, but that is not correct. To go from the first to the fifth floor, it's 4 floors Beyonce needs to walk. And from the first to the 9th is 8. So it will take her 5 times 2 = 10 minutes!

## A Winnie-the-Pooh problem

Winnie-the-Pooh, Piglet, Owl and Rabbit ate 70 strawberries, with everyone eating at least 1. They each ate a whole number of strawberries. Again, Pooh ate more than anybody else. Together, Piglet and Owl ate 45 strawberries. How many strawberries did Winnie-the-Pooh eat? What about Rabbit?

SOLUTION

If Owl and Piglet ate 45 together, and everyone ate at least 1, then Pooh and Rabbit ate 25 together. Since Rabbit ate at least 1, Pooh could eat at most 24. Since Piglet and Owl ate 45 together, they had to eat 23 and 22, because otherwise, one of them would eat the same number or more than Pooh.

Answer: Pooh ate 24 and Rabbit ate 1 strawberry. Poor Rabbit!

## A weighing problem

How can 2 kids and 1 adult cross a river in a row boat, if the boat can carry only 1 adult or up to 2 kids at a time?

SOLUTION

We will publish a solution to this puzzle on Wednesday, March 25th, at 9 pm or later. Each correct solution earns a an entry into a raffle to win a board game. Email your solution to info@cambridgemathcircle.org.