Maths Wizard Challenge

Here are the answers to this week’s tricky mathematical conundrums:

1)  In how many ways can the letters in the word PUPIL be arranged?

Answer: 60

Solution: If all five letters were different, there would be 5x4x3x2x1=120 arrangements. But the two Ps are identical, so 120 double-counts the arrangements. The required total is 60

2)  In how many of these arrangements are the two Ps next to each other?

Answer: 24

Solution: Treating the Ps as a single object, there are 4x3x2x1=24 arrangements in which the Ps are next to each other

Well done if you worked out the right answers! Come back next Monday for another pair of questions to get you thinking!

1) On a farm, the ratio of the number of pigs to number of sheep is 6:7 and the ratio of the number of sheep to number of cows is 3:8. In simplest form, what is the ratio of pigs to sheep to cows on the farm?

Answer: 18:21:56

Solution:  The ratio of pigs to sheep is 18:21 and the ratio of sheep to cows is 21:56. So the ratio of pigs to sheep to cows is 18:21:56

2) It is possible to make three different solid cuboids with 8 unit cube blocks, with dimensions of 1x1x8, 1x2x4 and 2x2x2 respectively (two cuboids are considered the same if they have the same shape). How many different solid cuboids can be made with 60 blocks?

Answer: 10

Solution: The possible dimensions are 1x1x60, 1x2x30, 1x3x20, 1x4x15, 1x5x12, 1x6x10, 2x2x15, 2x3x10, 2x5x6 and 3x4x5, so the required total is 10.

3) An ice-cream parlour sells 5 different flavours of ice-cream. Iona wants two scoops in a bowl. How many different flavour combinations can she order? (the flavours can be the same or different)

Answer: 15

Solution: Call the flavours A,B,C,D,E. There are 10 different combinations of two scoops with different flavours (AB, AC, AD, AE, BC, BD, BE, CD, CE, DE) and 5 different combinations with identical flavours (AA, BB, CC, DD, EE), so there are 15 combinations in total.

4) How many letters are in the answer to this question?

Answer: Four

Solutions: FOUR

5) At 10:15, Freya went for a short walk, then she painted until 11:00. She walked for one quarter of the time she painted. When did she start painting?

Answer: 10:24

Solution: Splitting 45 minutes into 5 equal parts, each part is worth 9 minutes. So she started painting at 10:24

6) A drawer contains 4 blue socks, 6 red socks and 8 purple socks. You randomly take socks from the drawer, arranging them into a pile. What is the smallest number of socks you must remove, to be certain that the pile contains a matching pair?

Answer: 4

Solution: If you take 3 socks, you might have 1 of each colour, but if you take 4 socks, it is impossible for them all to have different colours. So the answer is 4

7)  A drawer contains 4 blue socks, 6 red socks and 8 purple socks. You randomly take socks from the drawer, arranging them into a pile. What is the smallest number of socks you must remove, to be certain that the pile contains three different colours?

Answer: 15

Solution: If you take 14 socks, you might have 6 red socks and 8 purple socks, without any blue socks. But if you take 15 socks, it is impossible that they contain only 1 or 2 colours between them. So the answer is 15

8)  What is the fraction with lowest denominator, between 2/3 and 3/4?

Answer:  5/7

Solutions: No fractions with a denominator of 1, 2, 3, 4, 5 or 6 are between 2/3 and 3/4, but 5/7 is between 2/3 and 3/4. So the answer is 5/7

9)  Two numbers have a sum of -2 and a product of -1.25. What is their difference?

Answer: 3

Solution: The two numbers are 0.5 and -2.5, and their difference is 0.5-(-2.5)=3

10) What is the smallest positive integer divisible by each of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10?

Answer: 2520

Solution: 2x2x2x3x3x5x7=2520

11) How many different combinations of coins add to 7 pence?

Answer: 6

Solution: 7=5+2=5+1+1=2+2+2+1=2+2+1+1+1=2+1+1+1+1+1=1+1+1+1+1+1+1. There are six combinations overall.

12) In a shop, the price of a jumper is increased by a third. A week later, the price falls by a third. What fraction of the original price of the jumper is the current price?

Answer: 8/9

Solutions: After the price increase, the jumper’s price is 4/3 of the original price. This is equivalent to 12/9. So after the fall in price, the new price is 8/9 of the original price.

13)  In how many ways is it possible to pick three whole number angles to make an isosceles triangle?

Answer:  89

Solution: The base angles are equal, and can take any whole number value from 1 degree to 89 degrees, so this can be done in 89

14)  I have a circle made of paper. What is the smallest number of folds required to turn the circle into a polygon (a shape with straight edges)?

Answer: 3

Solutions:  Every edge requires a different fold, and every polygon has at least three sides, so the best we can hope for is three. And it is in fact possible to create a triangle with three folds; so the answer is three.

15)  Which negative number is four times as far away from 41 as from 2?

Answer:  -11

Solution: -11 (it differs from 2 by 13 and from 41 by 52)

16) What is the smallest cube divisible by 6 and 10?

Answer:  27000

Solution: The cube must be divisible by 2, 3 and 5. Further, in a cube, the exact power to which any prime appears in its prime factorisation must be a multiple of 3. So the smallest cube divisible by 6 and 10 is 2 cubed x 3 cubed x 5 cubed = 27000

17) Find three positive integers, each of which has an equal number of even factors and odd factors

Answer: Any number divisible by 2, except 4

Solution: Any number divisible by 2 but not by 4 will do. The first few examples are 2 (factors are 1,2), 6 (factors are 1, 2, 3, 6) and 10 (factors are 1, 2, 5, 10)

18) Find three positive integers, each of which has three times as many even factors as odd factors

Answer: Any number divisible by 8 but not by 16

Solution: Any number divisible by 8 but not by 16 will do. The first few examples are 8 (factors are 1, 2, 4, 8), 24 (factors are 1, 2, 3, 4, 6, 8, 12, 24) and 40 (factors are 1, 2, 4, 5, 8, 10, 20, 40)

19)  In how many ways can the letters in the word PUPIL be arranged?

Answer: 60

Solution: If all five letters were different, there would be 5x4x3x2x1=120 arrangements. But the two Ps are identical, so 120 double-counts the arrangements. The required total is 60

20)  In how many of these arrangements are the two Ps next to each other?

Answer: 24

Solution: Treating the Ps as a single object, there are 4x3x2x1=24 arrangements in which the Ps are next to each other

If you’d like to try some more tricky problem solving puzzles, here are some web sites you can look at. I’m certain you will find plenty to challenge you!

Click on any of the links to take you straight there.

Junior Maths Challenge

Here you can try online challenges from previous Junior Maths Challenges, as well as download past papers and solutions.

https://www.ukmt.org.uk/competitions/solo/junior-mathematical-challenge/archive

Nrich Website / Twitter

The NRICH Project aims to enrich the mathematical experiences of all learners. They have lots of activities, questions and games to develop your mathematical skills, whatever level you are. They definitely make maths fun!

https://nrich.maths.org/primary

They also post questions on twitter for all ages of students.

https://twitter.com/nrichmaths

Times Tables Rockstars

Times Tables Rock Stars is a carefully sequenced programme of fun daily times tables practice, which concentrates on a different times table each week. It has a small subscription of £7.20 a year for a family.

https://ttrockstars.com/home

As well as these web sites, I can also recommend the brilliant maths book Elastic Numbers, written by one of Hampton’s own Maths Teachers, Mr Griller.

Elastic Numbers is full of fun and challenging mathematical treats for the serious problem solvers among you! It’s available from most book shops and also online.