Maths Battle Sample Problems

These problems were used in a maths battle at Lutterworth College on July 4, 2013. The problems were kindly provided by Prof. Alexander Veselov.

Problem 1

In a large den there are hens and rabbits. They have 35 heads and 94 legs in total. How many rabbits are there?

Problem 2

What fraction of the rectangle PQRS is shaded?

Problem 3

A number of consecutive pages is missing from a book. If you add the page numbers of all the missing pages, you get 112. Which pages are missing?

Problem 4

Five brothers inherited three identical houses. They have divided up the estate fairly in the following way: each of the three older brothers got a house, but was obliged to pay 60 000 pounds in compensation. The total compensation from the three older brothers was shared equally between the two younger brothers. What was the price of one house?

Problem 5

Find the smallest 3-digit number, which is neither prime nor divisible by 2, 3 or 5.

Problem 6

A man has 2 credit cards, with a 4-digit pin code for each of the cards. The first code is 4 times larger than the second code, and also is the reverse of it (for example, 5134 is the reverse of 4315). What are the pin codes for the credit cards?

Problem 7

Figure A and Figure B represent the path of a fish in an aquarium when looking at it from the front and from the right hand side respectively.

Draw the path of the fish when looking at the aquarium from the top.

Problem 8

Jo has written all the natural numbers between 1 and 1000 in a row:
12345678910111213141516171819202122………………9991000.
What is the 1005-th digit in this row?

Problem 9

Replace the letters by digits so that the following calculation is correct:

   CROSS
+ CROSS
   ----------
   SPORT

Different letters must correspond to different digits.

Problem 10

Three fortune-tellers sit in a row: The Truth (who always says the truth), The Lie (who never says the truth), and The Cunning (who sometimes tells the truth, and sometimes lies). A philosopher arrives to decide who is who. He asks the person sitting on the left: “Who is sitting next to you?” The answer is: “The Truth.” Then he asks the person sitting in the middle: “Who are you?” The answer is: “The Cunning.” Finally, he asks the person sitting on the right: “Who is sitting next to you?” The answer is: “The Lie.” Can you tell who is sitting where?