Coins
A man has 5 coins in his pocket. Given the clues, can you work out
what the coins are?
Age
11 to 14
Challenge level
Problem
A man has 5 coins in his pocket.
- He can make 13 different amounts of money with his coins
- The amounts of money he can make end with one of two possible digits.
- He cannot make up exactly 70 pence
- He cannot afford an item costing 1 pound
- He can make a prime number bigger than 10 with his coins.
Given these clues, can you work out what the coins are?
Is there only one possible combination of coins?
Extension:
Remove one of the clues. Are there any new possibilities for the values of the 5 coins?
Replace that clue and remove a different one, work out the new possibilities. Repeat for the other clues.
Can you remove a clue and only end up with your original solution(s)?
Which clues have the biggest impact?
Getting Started
British coins in current circulation are:
1p (penny)
2p
5p
10p
20p
50p
£1 (=100p)
£2
Start by considering individual clues and how they restrict the possibilities.
Student Solutions
Class 9Y-1 from Downend School sent us the correct solution and showed that it satisfied the criteria:
The five coins are 1p, 10p, 10p, 20p, 20p.The thirteen amounts are 1p, 10p, 11p, 20p, 21p, 30p, 31p, 40p, 41p, 50p, 51p, 60p, 61p.
These amounts only end in the digits 0 or 1.
11p, 31p, 41p and 61p are all prime amounts of money.
Similar solutions were sent by Kate and Hannah from Marist College in New Zealand; students from Wingham Primary School; James from Wellington School; Can, Jack and Lee from Ashmount Primary School; Patrick and Natalie also from Ashmount Primary School; Thomas, Jack, Jim, Jenny and Timothy from Bay House School; Tasmin from St. Pauls; Kirsty from Herts and Essex High School; Levi from Greenwood Junior School; Tommy from St Mary's Hall, Lincoln and Jaehyung from ACS Egham International School, and Daniel from Ysgol Dyffryn Dulas Corris.
Jae explained his reasoning:
The first clue I used was: He can not make exactly 70p.With that clue, I knew that for example it couldn't be 10, 10, 50, 5, 20 or 20, 20, 20, 10, 1
Then I looked at the clue The amounts of money he can make end with one of two possible digits.
That meant in the ones, it could be 5 and 0 or 1 and 0.
The last clue I used was He can make 13 different amounts of money with his coins.
I used trial and error and I got 1p, 10p, 10p, 20p and 20p.
It couldn't make exactly 70p and it was under £1 as well.
1, 10, 11, 20, 21, 30, 31, 40, 41, 50, 51, 60, 61 was the combinations he could make. That was 13 combinations all together.
And if you added all of the coins together, it would be 61p, which is a prime number.
Also there was only two digits in the ones place, 1 and 0. That is how I got my answer.
Alex from Bristol Grammar School also explained his reasoning:
A 1p coin, two10p coins, two 20p coins. A 1p coin is needed to make the prime number above 10 and the rest must be mutliples of 10 so that the final digit does not change. The 10 and 20 p coins were chosen through system of elimination.Sammy & Lucy from Hove Park School also showed that their solution satisfied all the criteria:
A Man has 5 Coins and his coins are 10p, 10p, 20p, 20p, 1pThe specifications are:
He can make 13 different amounts of money with his coins.
The amounts he can make are.... 21p, 11p, 10p, 20p, 1p, 30p, 40p, 50p, 60p, 31p, 41p, 51p, 61p Which totals to 13 amounts.
The amounts of money he can make end with one of two possible digits.
All of the amounts above ended with either a 0 or a 1.
He cannot make up exactly 70 pence.
70p is not one of the totals seeing as 61 is the highest.
He cannot afford an item costing £1.
£1 cannot be one of the totals seeing as 61p is the highest.
He can make a prime number bigger than 10 with his coins.
61 is a prime number and over 10.
Well done to you all.
Teachers' Resources
Why do this problem?
This problem requires students to work systematically and apply some understanding of primes. It could also be adapted as a basis for work familiarising students with UK coinage.Possible approaches
This task could be a suitable homework or "Problem of the Week" during a topic on Probability, Money or Primes. Solutions to the extension questions could be posted on a display board where students could check them and contribute further ideas over an extended period of time.Alternatively the task could be used as an exercise in group work - give one member of the group a pen and paper, and give each of the others one (or two) clues. They must not show the clues to anyone or write them down. They can read clues out or explain them to the group. Each student is responsible that their clues are satisfied. The group needs to work together to find a solution to all the clues.
Key questions
Can you choose a good clue to start working on, one that gives you enough information to get started?Are you considering ALL possibilities, or just trying to spot something that works?
Possible extension
This could be a good time to talk about the phrase "necessary
and sufficient". Ask students to devise a set of clues that define
a unique set of coins, and where no individual clue can be deleted
without losing the uniqueness of the solution.
Possible support
Students could be given a pile of coins, (or a worksheet
ofphotocopied/drawn coins) and asked to select sets of 5 coins from
these, satisfying suitable clues - the total is...
an even number
one more than a multiple of 7
less than 10
all the coins are different, etc.
They could be asked to devise clues for each other, or to look
for combinations of clues that define one specific set of
coins.