5917
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1
2011-02-01T00:00:01
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A man has 5 coins in his pocket.<br></br>
<br></br>
<ul>
<li>He can make 13 different amounts of money with his coins</li>
</ul>
<ul>
<li>The amounts of money he can make end with one of two possible digits.</li>
</ul>
<ul>
<li>He cannot make up exactly 70 pence</li>
</ul>
<ul>
<li>He cannot afford an item costing 1 pound</li>
</ul>
<ul>
<li>He can make a prime number bigger than 10 with his coins.</li>
</ul>
<br></br>
Given these clues, can you work out what the coins are?<br></br>
Is there only one possible combination of coins?<br></br>
<br></br>
<h3>Extension:</h3>
<div>Remove one of the clues. Are there any new possibilities for the values of the 5 coins?</div>
<div>Replace that clue and remove a different one, work out the new possibilities. Repeat for the other clues.</div>
<br></br>
<div>Can you remove a clue and only end up with your original solution(s)?</div>
<br></br>
<div>Which clues have the biggest impact?</div>
<br></br>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=6469&part=">Click here for a poster of this problem</a>.</div>
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<p class="editorial">Class 9Y-1 from Downend School sent us the
correct solution and showed that it satisfied the criteria:</p>
The five coins are 1p, 10p, 10p, 20p, 20p.<br></br>
The thirteen amounts are 1p, 10p, 11p, 20p, 21p, 30p, 31p, 40p,
41p, 50p, 51p, 60p, 61p. <br></br>
These amounts only end in the digits 0 or 1. <br></br>
11p, 31p, 41p and 61p are all prime amounts of money.<br></br>
<p class="editorial">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 and Lincoln and Jaehyung from ACS Egham International
School.</p>
<p class="editorial">Jae explained his reasoning:</p>
The first clue I used was: He can not make exactly 70p.<br></br>
With that clue, I knew that for example it couldn't be 10, 10, 50,
5, 20 or 20, 20, 20, 10, 1 <br></br>
<br></br>
Then I looked at the clue The amounts of money he can make end with
one of two possible digits. <br></br>
That meant in the ones, it could be 5 and 0 or 1 and 0. <br></br>
<br></br>
The last clue I used was He can make 13 different amounts of money
with his coins. <br></br>
I used trial and error and I got 1p, 10p, 10p, 20p and 20p.<br></br>
<br></br>
It couldn't make exactly 70p and it was under £1 as
well. <br></br>
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.
<br></br>
<br></br>
And if you added all of the coins together, it would be 61p, which
is a prime number. <br></br>
Also there was only two digits in the ones place, 1 and 0. That is
how I got my answer. <br></br>
<p class="editorial">Alex from Bristol Grammar School also
explained his reasoning:</p>
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.<br></br>
<p class="editorial">Sammy & Lucy from Hove Park School also
showed that their solution satisfied all the criteria:</p>
A Man has 5 Coins and his coins are 10p, 10p, 20p, 20p, 1p<br></br>
<br></br>
The specifications are: <br></br>
He can make 13 different amounts of money with his coins.<br></br>
The amounts he can make are.... 21p, 11p, 10p, 20p, 1p, 30p, 40p,
50p, 60p, 31p, 41p, 51p, 61p Which totals to 13 amounts.<br></br>
<br></br>
The amounts of money he can make end with one of two possible
digits. <br></br>
All of the amounts above ended with either a 0 or a 1. <br></br>
<br></br>
He cannot make up exactly 70 pence.<br></br>
70p is not one of the totals seeing as 61 is the highest. <br></br>
<br></br>
He cannot afford an item costing £1. <br></br>
£1 cannot be one of the totals seeing as 61p is the
highest. <br></br>
<br></br>
He can make a prime number bigger than 10 with his coins. <br></br>
61 is a prime number and over 10.<br></br>
<p><span class="editorial">Well done to you all.</span></p>
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<h3>Why do this problem?</h3>
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. <br></br>
<h3>Possible approaches</h3>
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.<br></br>
<br></br>
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 <span style="text-decoration: underline;">can</span> 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.<br></br>
<h3>Key questions</h3>
Can you choose a good clue to start working on, one that gives you
enough information to get started?<br></br>
Are you considering ALL possibilities, or just trying to spot
something that works?<br></br>
<h3>Possible extension</h3>
<div>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.</div>
<h3>Possible support</h3>
<div>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...</div>
<div>an even number</div>
<div>one more than a multiple of 7</div>
<div>less than 10</div>
<div>all the coins are different, etc.</div>
<br></br>
<div>They could be asked to devise clues for each other, or to look
for combinations of clues that define one specific set of
coins.</div>
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British coins in current circulation are:<br></br>
1p (penny)<br></br>
2p<br></br>
5p<br></br>
10p<br></br>
20p<br></br>
50p<br></br>
£1 (=100p)<br></br>
£2<br></br>
<br></br>
Start by considering individual clues and how they restrict the
possibilities. <br></br></mdoxml>
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<mdoxml version="1.0">1, 10, 10, 20, 20 <br></br></mdoxml>
2
5
0
0
1
0
0
Coins
A man has 5 coins in his pocket. Given the clues, can you work out
what the coins are?
Decision Mathematics and Combinatorics
Combinations
Using, Applying and Reasoning about Mathematics
Working systematically
Numbers and the Number System
Prime numbers
Decision Mathematics and Combinatorics
Logic