Maths Wizard Challenge

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

1) What is the smallest positive integer whose digits have a product of 60?

Answer: 256

Solution: There are no such integers with one or two digits. The required number is 256.

2) Alice has twice as many marbles as Bob. Then Alice gives 10 of her marbles to Bob. Now Bob has three times as many marbles as Alice. How many marbles do they have in total?

Answer: 24

Solution: Alice and Bob had 16 and 8 marbles respectively before the switch, and 6 and 18 marbles respectively after the switch. In total they have 24

3) Two fair dice are rolled and the two numbers obtained are added. What is the probability that this sum is divisible by 5?

Answer: 7/36

Solution: There are 6×6=36 possible outcomes, of which 7 produce a sum divisible by 5 (1 and 4, 2 and 3, 3 and 2, 4 and 1, 4 and 6, 5 and 5, 6 and 4). So the answer is 7/36.

4)) A triangular prism has V vertices, E edges and F faces. What is the value of V+E+F?

Answer: 20

Solution: V=6, E=9 and F=5, so V+E+F=6+9+5=20.

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

1)   In a sale, price are reduced by 25% (a quarter). One of the shop managers mistakenly reduces the price of a t-shirt twice instead of just once. The new price is £27. What was the original price?

Answer: £48

Solution: Before the second reduction, the price was £36. So before the first reduction, the price was £48.

2) What decimal is halfway between 3.75 and 5.2?

Answer: 4.475

Solution: 3.75+5.2=8.95, and half of 8.95 is 4.475.

3) In a triangle, the smallest angle is 5 degrees smaller than the middle angle, which is 20 degrees smaller than the largest angle. What is the size of the smallest angle?

Answer: 50 degrees

Solution: The three angles are 50 degrees, 55 degrees and 75 degrees, of which 50 degrees is the smallest.

4) What is the sum of all the multiples of four between 10 and 1010?

Answer: 127500

Solution: There are 250 multiples of four between 10 and 1010, from 12 to 1008 inclusive. These can be paired up into 125 pairs, so that the sum of each pair is 1020 (12+1008, 16+1004 and so on…). So the grand total is 125×1020=102000+20400+5100= 127500

5)   A pair of gloves costs £5.50 more than a hat. Together the gloves and the hat cost £18. What is the price of the hat?

Answer: £6.25

Solution: The hat costs £6.25 (the gloves cost £11.75)

6)   In a football tournament, each of 15 teams plays every other team exactly once. How many matches are there in total?

Answer: 105

Solution: Each of the 15 teams plays 14 matches, suggesting 15×14=210 matches in total. But this counts every match twice (for example, team A v team B and team B v team A). So the final total is 210/2=105

7)   A collection of 1×1 unit squares is used to construct a rectangle of area 40. What is its smallest possible perimeter?

Answer: 26

Solution: The perimeter will be minimised when the side lengths are as close together as possible. This happens with side lengths of 5 and 8, so the answer is 2x(5+8)=26

8)  What is the largest three-digit number divisible by 4, 6 and 8?

Answer: 984

Solution: A number is divisible by 4, 6 and 8 precisely when it is divisible by 24, and the largest three-digit multiple of 24 is 984.

9) What is the surface area of a cube with volume 64?

Answer: 96

Solution: The length of one edge is 4, so the area of one face is 16, hence the surface area is 16×6=96.

10) Mary has read two fifths of a novel, and has 171 pages left. How many pages are in the book?

Answer: 285

Solution: 171 pages represents three fifths of the book. So one fifth is 171/3=57 pages, and five fifths (that is, the whole book) is 57×5=285 pages.

11) The sum of five consecutive integers is 5. What is the sum of their squares?

Answer: 15

Solution: The five integers are -1,0,1,2 and 3. So the sum of their squares is 1+0+1+4+9=15.

12) Thomas begins facing northwest. He then turns clockwise through 4050 degrees. In what direction is he now facing?

Answer: northeast

Solution: 3600 degrees is 10 full revolutions. Another full revolution completes 3960 degrees, and 4050 degrees is only 90 degrees further. So he is now facing northeast

13)  A fair 6-sided die is rolled twice. What is the probability that the second number is higher than the first?

Answer: 5/12

Solution: There are 15 combinations that result in the second number higher then the first, out of 36 combinations in total, so the required probability is 15/36=5/12

14)  What number is three quarters of four fifths of five sixths of 170?

Answer: 85

Solution: By cancelling common factors, we find that 3/4 x 4/5 x 5/6 = 3/6 = 1/2 , and half of 170 is 85

15)  The sum of the interior angles of an n-sided polygon is twice the sum of the interior angles of a hexagon. What is the value of n?

Answer: n=10

Solution: The interior angles in a hexagon sum to 4×180 degrees. So the n-sided polygon has an interior angle sum of 8×180 degrees. Hence n=10

16)  Rachel walks 2 miles from her house to the park at a speed of 4 miles per hour. She runs back home at a speed of 10 miles per hour. What is her average speed for the whole journey?

Answer: 5 5/7 miles per hour

Solution: It takes Rachel half an hour to walk to the park, and one fifth of an hour to return. That is 1/2 + 1/5 = 5/10 + 2/10 = 7/10 of an hour in total, to cover 4 miles. So her average speed is 4 / 0.7 = 40/7 = 5 5/7 miles per hour

17) What is the sum of the lengths of the edges of a 3x4x5 cuboid?

Answer: 48

Solution: There are four edges of each of the lengths 3, 4 and 5, so the required sum is 4(3+4+5)=48

18) Two siblings have ages in the ratio 2:3, but in three years the ratio will be 7:10. What is the sum of their ages now?

Answer: 45

Solution: Their current ages are 18 and 27 (in three years their ages will be 21 and 30), so the sum of their current ages is 45

19) What number falls by 12 when tripled?

Answer: 6

Solution:If the number is x, then we have 3x=x-12, so 2x=-12 and x=-6

20) Find all pairs of different positive integers, each less than 40, whose lowest common multiple is 40.

Answer: (5,8), (8,10) and (8,20)

Solution: Both integers in the pair must be factors of 40, one must be divisible by 5, and the other must be divisible by 8. The required pairs are (5,8), (8,10) and (8,20)

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.