### Question 1

A set of real numbers: 4, x, 1, 9, 6 has the property that the median of the set of numbers is equal to the mean. What values can x be?

### Question 2

My digital watch uses ‘12 hour clock’ and shows hours and minutes only. For what fraction of a complete day is at least one "1" showing on the display?

### Question 3

How many rectangles (including squares) can be drawn using the squares on a 4x4 chessboard? Rotations and translations are permitted.

### Question 4

This network has nine edges which meet at six nodes. The numbers 1,2,3,4,5,6 are placed at the nodes, with a different number at each node.

### Question 4 (cont)

Is it possible to do this so that the sum of the two numbers at the ends of an edge is different for each edge? Either show a way of doing this, or prove that it is impossible.

This problem taken from http://nrich.maths.org/weekly

### Question 5

A six digit positive integer has the property that when its digits are rearranged, the new number is double the original. What is the smallest number this can be?

### Question 6

2013 can be written as the sum of consecutive positive integers, all of which have two digits. What are the first and last two digit numbers in the sum?

### Question 7

Temperature can be measured in degrees Fahrenheit or Celsius. What temperature is represented by the same number on both scales?

### Question 8

2/3 of men answering this question got it right and 3/4 of women answering it got it right. The same number of men and women were correct. What fraction of all respondents were correct?

### Question 9

16 is a special number because when it is written as numbers with indices it can be written as either 2 4 or 4 2 .

What is the next number that can be written as both x y and y x ?

= (where x and y are both positive integers)

1 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 0 1 0 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0

### Question 10

0 0 1 0 0 1 0 1 1 0 1 0 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 0 1 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 1 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1

### Question 11

London has longitude 0 °. Cardiff has longitude 3 ° W. They have the same latitude.

If the sun rose at 4:40am this morning in London, at what time did it rise in Cardiff?

### Question 12

In the game fizz-buzz, the positive integers are called out in order starting from 1 but replacing any number which is a multiple of 3 with ‘fizz’ and replacing any number which is a multiple of 5 with ‘buzz’. So the game begins: 1, 2, fizz, 4, buzz, fizz, 7, 8, fizz, buzz, 11, fizz, 13, 14, fizzbuzz , 16, 17… What is the 2013th number to be said aloud?

### Question 13

Joining adjacent midpoints of the edges of a regular tetrahedron creates a regular octahedron. What is the ratio of the volume of the octahedron to the volume of the original tetrahedron?

### Question 14

A regular octagon has side length 2. What is its area?

### Question 15

2013! ends in a string of zeros. How many of them are there?

Teacher notes • These questions formed the QR maths quiz at the MEI conference 2013. • They are in approximate order of difficulty … although difficulty is in the mind of the solver!

• You might want to use a selection of these questions as a quiz for your own students. Most questions are accessible to KS4 students and younger.

• Question 9 has an alternative ‘visual’ version, please see the attached Excel sheet.

• Answers on the next page

2.

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7.

0, 5, 10 ½ 100 Impossible. Sum of edges has to be 63 (sum of 3 to 11) whereas each node is even, hence each of the 1-6 digits will be used 2 or 4 times each. This will give an even total.

142 857 and 285714 45 and 77 -40 8.

9.

12/17 No other positive integer solutions exist 10. 13 11. 4:52am 12. 3773 13.

14.

½ 8+8√2 15. 501