# Maths Wizard Challenge

## This Week’s Maths Wizard Problem…

**Hello young Maths Wizards!**

**Our magical leaderboards are back!**

**If you worked out the answers to last week’s questions and submitted your answers, do check the leaderboards to see if your answers were correct and you have made it onto our new leaderboards. **

**Five points are awarded for each correct answer, so there are 10 possible points to be won each week. There will be one leaderboard each for Years 4, 5 and 6, and as soon as you are awarded points your first name will be added. **

**Remember to check back each week to see how you are doing.**

**Don’t forget, if you’d like some extra problem solving questions, check out the Further Resources tab, which includes some fun maths websites for you to try.**

**Click on my wand to see this week’s wizarding maths problems**.

**Good Luck!**

**21 September 2020**

**We have two new questions for you to test your mathematical thinking this week**

*Check back next week to see if your answers are correct!*

- In a triangle, the interior angles are in the ratio 3:4:5. What is the size of the largest angle?

- What is the size of each interior angle in a regular 100-sided polygon?

Please submit your answers below and if correct you’ll name will be added to the leader board next week.

**Here are the answers to last week’s mathematical conundrums:**

1. Over a period of four days, Ellie eats several strawberries. Each day after the first day, she eats twice as many strawberries as she did the day before. In total she eats 90 strawberries.

How many strawberries did she eat on the last day?

* Answer:* 48

** Solution:** Ellie ate 6, then 12, then 24, then 48 strawberries across the four days. So she ate

**48**on the last day.

2. Over a period of four days, Taran eats lots of peas. Each day after the first day, he eats two thirds as many peas as he did the previous day. In total he eats 325 peas.

How many peas did he eat on the first day?

** Answer:** 135

** Solution:** Taran ate 135, then 90, then 60, then 40 peas across the four days. So he ate

**135**on the first day.

**Well done if you worked out the right answers!**

**Maths Wizard Leaderboards**

Congratulations to everyone who correctly worked out the answers to my mathematical conundrums. Five points are awarded for each correct answer, so there are 10 possible points available to be won each week.

Here are the new Leaderboards for the Autumn Term for Years 4, 5 and 6.

Can you see your name?

**Year 4**

**Year 5**

**Year 6**

1) The diagram below shows a partially completed magic square in which, all rows, columns and diagonals add up to the same number.

Work out the value of X and y.

*Answer: 13*

2) **Without using a calculator**, work out 123123123123 ÷ 123

*Answe**r: **1001001001*

3) Molly cuts a cake into 10 large pieces. She then takes half of these and cuts them into 3 medium pieces each.

Finally, she takes a third of the medium pieces and cuts each of them into 2 small pieces.

(i) How many pieces of cake are there at the end?

*Answer: 25 pieces*

* Solution:* Of the 10 large pieces, 5 are left alone. The other 5 are cut into 15 medium pieces. Of these 15 pieces, 10 are left alone. The other 5 are cut into 10 small pieces. So at the end, there are 5 large, 10 medium and 10 small slices; that is

**25**pieces in total.

(ii) Barney takes one piece of each size. What fraction of the original cake does this represent?

*Answer: 1/6 (one sixth)*

* Solution: * Each large piece represents 1/10 of the original cake. Each medium piece represents 1/30 of the original cake. Each small piece represents 1/60 of the original cake. So the fraction represented by Barney’s portion is (1/10)+(1/30)+(1/60)=(6/60)+(2/60)+(1/60)=(10/60)=

**1/6**(

**One sixth**).

4) Kam starts listing the positive whole numbers that only use the digits 0 or 1. The list begins 1, 10, 11, … . What is the 10th number in the list?

* Answer:* 1010

* Solution:* The first 10 numbers are 1, 10, 11, 100, 101, 110, 111, 1000, 1001 and 1010. So the 10th number is

**1010**.

5) Shram starts listing the positive whole numbers that only use the digits 2 or 3. The list begins 2, 3, 22, … . What is the 23rd number in the list?

* Answer:* 3222

** Solution**: There are two one-digit numbers, four two-digit numbers and eight three-digit numbers in the list, the last of which is 333. This is the 14th number in the list. The next nine numbers are 2222, 2223, 2232, 2233, 2322, 2323, 2332, 2333 and 3222. So the 23rd number is

**3222**.

6) What is the largest two-digit number that can be written as a sum of five consecutive whole numbers?

** Answer:** Any multiple of 5 will work, and the largest is 17+18+19+20+21=

**95**.

7) What is the largest two-digit number that can be written as a sum of five consecutive whole numbers, and a sum of six consecutive whole numbers?

** Answer:** We are looking for a multiple of 5 that is also an odd multiple of 3. The largest is 13+14+15+16+17=10+11+12+13+14+15=

**75**.

8) In how many ways can the squares in a 2×2 grid of squares be coloured red, green or blue, so that all three colours are used, but no squares sharing an edge have the same colour?

* Answer:* 12

** Solution:** Either the top-left and bottom-right squares are the same colour, or the top-right and bottom-left squares are the same colour. In either case, there are 3 ways to colour the pair, then 2 ways to colour the two remaining squares. So each case produces 6 possible colourings, or

**12**in total.

9) How may five-digit numbers contain exactly two different digits?

* Answer:* 1215

* Solution:* There are 9 choices for the first digit, then 9 choices for the value of the other digit that appears, then 2 ways to choose the second digit, 2 ways to choose the third, 2 ways to choose the fourth, and 2 ways to choose the fifth and final digit. That is 2x2x2x2=16 ways to choose the last four digits; except that 1 of these choices will produce a number where every digit is the same (like 44444). So there are only 15 ways to choose the last four digits, and the final total is 9x9x15=

**1215**.

10) Three kittens and two puppies weigh 17kg, while two kittens and three puppies weigh 18kg.

How much will one kitten and one puppy weigh?

**Answer: 7kg**

11) The mean length of a crocodile is 8.5 m and the mean length of an alligator is 4 m. The Maths Wizard has 4 crocodiles and 6 alligators.

What is the mean length of all 10 of these creatures?

*Answer: 5.8m*

12) The Maths Wizard has a large pile of delicious chocolates.

On Monday the Maths Wizard ate half of the chocolates.

On Tuesday he ate one third of what remained.

On Wednesday he ate one quarter of the rest.

And on Thursday he ate one fifth of the much diminished remaining pile of delicious chocolates.

What fraction of the original pile of delicious chocolates now remain?

*Answer: 1/5*

13) How many minutes are there from 11:11 until 23:23 on the same day?

**Answer: 732**

14) Alice thinks of a number. She doubles it, then subtracts 7, then halves the result, ending up with -2. What number did she start with?

* Answer:* 1.5

* Solution:* Alice starts with 1.5. She doubles it to get 3, subtracts 7 to get -4, then halves this to end up with -2.

15) Bob thinks of a number. He cubes it, then adds 30, then multiplies the result by -1, ending up with the number he first thought of. What is this number?

**Answer: -3**

* Solution:* Bob starts with -3. He cubes it to get -27, adds 30 to get 3, then multiplies this by -1 to get back to -3.

**Why not take a look at some more tricky problem solving puzzles at the following websites. Just click on the link and it will take you straight there.**

**Junior Maths Challenge**

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

**Nrich Website / Twitter** (on twitter they post questions)

https://nrich.maths.org/primary

https://twitter.com/nrichmaths

**Times Tables Rockstars** (great fun reinforcing times table skills, for a small subscription of £6 pa)

**You may also like to take a look at Elastic Numbers, written by one of Hampton’s own Maths Teachers, Mr Griller.**

Available from most book shops and also online, *Elastic Numbers* is full of fun and challenging mathematical treats for the serious problem solvers among you!