Durham maths mysteries

“Putting a little mystery
into mathematics”

November 2005
Durham LA

The Mysteries

Index of mathematical mysteries and details of the
mathematics needed to solve each of them:

Ratio and Proportion

Fractions and percentages of quantities; equivalence of fractions and
percentages; multiples of ratios; average of two percentages.

Directed Numbers

Addition and subtraction of directed numbers, odd and even numbers
(integers).

Algebra

Addition and subtraction of linear algebraic terms of the form aϰ + b;
expansion of c(aϰ + b) and factorisation of acϰ + bc.

Properties of Shape

Lines of symmetry; centres of rotation; properties of quadrilaterals;
‘regular’ shapes.
A set of 15 shape cards are available to provide visual support.

Locus

Constructions including right angles, parallel lines, perpendicular
bisector of a line; loci of points which are equidistant from either a
fixed point or a fixed line.

Probability

Probability line 0-1; knowledge of probability of certainty; vocabulary:
‘evens chance’; likelihood and chance.

durhammathsmysteries-120711082030-phpapp02.doc 1 / 16

LiL ‘RATIO AND PROPORTION’ MYSTERY

Who should get the maths prize?

You may wish to draw up a table to show each pupil’s marks and percentages on
each paper and their average percentage over both papers. Hence, put the pupils
in rank order.

Adam got 3/4 of the
Brian scored 10
marks on Paper 1
The maths exam more marks on
but only 30% on
consists of 2 papers. Paper 1 than he did
Paper 2 when he
on Paper 2.
forgot his calculator.

Cathy scored 62
Susie’s average
marks on Paper 1
percentage on the Brian got 2/5 of the
and got 60 marks
two papers was marks for Paper 2.
more than Adam on
65%.
Paper 2.

The ratio of Susie’s percentage
Debbie’s mark on
Debbie’s marks on on Paper 1 was
Paper 1 was 10 less
Paper 1 to her 15% higher than
than Brian’s on
marks on Paper 2 Adam’s on the same
Paper 1.
was 5:4. paper.

There are 100 Pupils are ranked by
marks available on their average
Paper 1 and 150 on percentage on the
Paper 2. two papers.

EXTENSION

Would the order change if pupils were ranked on total marks scored?


LiL ‘DIRECTED NUMBERS’ MYSTERY

There is a 3 x 3 grid of integers with one number in each grid square.

Use the cards to decide which number lies in each grid square.

No number is bigger
There are two equal There are two equal
than 6 or smaller
positive numbers. negative numbers.
than -5.

The largest number The sum of the The sum of the
is in the centre totals of the three totals of the three
square. rows is zero. columns is zero.

The middle row The top row adds
The difference
adds up to the up to the same total
between the largest
same total as the as the diagonal from
and smallest
diagonal from top top left to bottom
numbers is 11.
right to bottom left. right.

The difference
There are an equal The right hand
between the number
number of positive column contains two
in the centre square
and negative numbers the same
and that in the
numbers. bottom right is 8. and adds up to -3.

The two 3s do not The bottom row Numbers in the top
lie in the same row contains no odd row are all different
or column. numbers. and are all odd.

The smallest
The smallest
number is opposite
number occupies a
one of the two equal
corner square.
negative numbers.


LiL ‘ALGEBRA’ MYSTERY

There is a 3 x 3 grid square with one linear algebraic expression of the form
aϰ + b in each square.

Use the cards to decide which expression lies in which square.

The expression in The sum of the
The right hand the middle of the diagonal top left to
column adds up to bottom row is twice bottom right is 3 less
5ϰ + 1. that in the top left than that on the
corner. other diagonal.

Two of the cards in The sum of the
middle row is 8 times
The bottom row the right hand
that of the card in the
adds up to 5ϰ + 16. column add up to right hand column of
2ϰ + 1. that row.

The sum of the The cards in the top The sum of the
expressions in the left
right and bottom left middle column is 10
hand column is 4 times that of the card
times the expression
squares are the only
in the right hand
in the bottom right cards which do not
column of the middle
hand corner. have two terms. row.

The difference The sum of the top
The sum of the
between the top two row is 3 times that
middle column is
cards in the left of the left hand card
10ϰ - 10.
hand column is 6. in the middle row.

The expression on
the middle card is
4ϰ - 3 more than
that to its right.


LiL ‘PROPERTIES OF SHAPE’ MYSTERY

There is a 3 x 3 grid with one shape drawn in each grid square.
Use the cards to decide which shape is in which square.
Is your answer unique?

The shape to the left
The shape in the top There is a square of the small square
left hand corner has directly above the has 2 lines of
3 lines of symmetry. hexagon. symmetry and 4
right angles.

Each of the shapes
Two shapes each Each row and in the top right and
have 4 lines of column contains 2 bottom left hand
symmetry. quadrilaterals. corners has one line
of symmetry.

5 of the
One shape has no
quadrilaterals 4 of the shapes are
straight sides and
include at least one regular with straight
one centre of
pair of parallel sides.
rotation.
sides.

One of the shapes
5 shapes have all 4 quadrilaterals
has one line of
sides equal in have diagonals
symmetry and its
length. which cut at 90º.
diagonals cut at 90º.

Shapes in the middle
Shapes in the
row and in the right
middle column No shape has more
hand column contain
contain a total of 14 than 6 vertices.
an infinity of lines of
lines of symmetry.
symmetry.

EXTENSION

(i) Create an additional card to make your solution unique.
(ii) Replace one of the cards with one of your own. Does your new problem have a
solution? Is it unique?
(iii) Design your own 3 x 3 shape grid and a set of cards.


LiL ‘LOCUS’ MYSTERY

You have the plan of an area of land centred on a four sided field reputed once to
have belonged to a notorious highwayman.
Use the cards to construct the diagram and hence solve the puzzle of where to
dig for the treasure.

The right angle in
CD is the longest quadrilateral ABCD Point E is a quarter
side. is opposite the way along DC.
longest side.

Points L and M lie
on the line through
AF is parallel to DC. A and F and are F lies on BC.
each 4cm from point
E.

PQ bisects EF at R. AB is 1½ times AD. BC is 9cm long.

The line through D
Angle BAD is
AD is 3cm long. and A meets PQ at
obtuse.
T.

The point where the The treasure is
Point L is closer to A treasure is buried
closer to D than to
than is point M. lies within 8cm of
point B. C.

The treasure lies The treasure is
inside quadrilateral buried at a labelled CE is 3” long.
ABCD. point.

EXTENSION
Investigate what happens as you allow the length of BC to vary.


LiL ‘PROBABILITY’ MYSTERY

0 0.1 0.5 1

Six friends enter a race. Use the following cards to determine who is most likely
to win the race and with what probability. In what sequence would you expect the
runners to finish the race?

The probability that
A wins is equal to Two runners have a
C is twice as likely
the sum of the better than evens
to win as B. probabilities that F chance of winning.
or C win.

Two runners have The probability that The probability that
an equal but not C wins is half the D wins is half that of
very good chance of combined probability each of two other
winning. that D or E win. runners.

The least likely
The chance that C Runners B, F and A
winner has a
wins is less likely have a combined probability 0.6
than two other probability equal to smaller than the
runners. that of certainty. most likely winner.

Runner F has a Each runner’s
Runner A is three
probability of probability of
times more likely to
winning that is 1/3 winning is a multiple
win than runner B.
that of runner A. of 0.1.

Runners A and C Only one runner has
have a combined a chance of winning
probability of 1. greater than 2/3.

EXTENSION
What is the smallest number of cards that you need to solve the problem? Which
cards do you need?


-5 3 1

3 6 -2

0 -4 -2


2ϰ + 3 ϰ - 12 3ϰ

2ϰ - 3 5ϰ - 4 ϰ-1

8 4ϰ + 6 ϰ+2


ϰ - 10 6 5ϰ - 13


TEACHERS’ NOTES

Ratio and Proportion

Paper 1 Paper 2 Average Rank by Total Rank by
Pupil mark % mark % % average % mark total
number
Adam 75 75 45 30 52.5% 4 120 4
Brian 70 70 60 40 55% 3 130 3
Cathy 62 62 105 70 66% 1 167 1
Debbie 60 60 48 32 46% 5 108 5
Susie 90 90 60 40 65% 2 150 2

Directed Numbers

Support could be to provide pupils with the set of integers involved ie:

-5 -4 -2 -2 0 1 3 3 6

Solution:
-5 3 1

3 6 -2

0 -4 -2

Algebra

Support could be to provide pupils with the set of 9 algebra cards (page 10/16) that form
the solution or the set of 12 cards, which include some ‘rogue’ cards.

Properties of Shape

Support could be to provide pupils with the set of 9 shape cards (page 12/16) that form
the solutions or the set of 15 cards, which include some ‘rogue’ cards.

Solution: see page 12/16 for one solution. Another solution is to swap the kite with the
isosceles trapezium.


Locus

Demonstrate the range of possible solutions as BC varies by using a dynamic geometry
package.

Probability

More cards are given than are necessary to find a solution - a smaller sufficient set
might support some pupils.

Solution: E is most likely to win, with a probability of 0.7. Sequence is E (first), A, C, B
or F in either order, D (last).


Durham maths mysteries

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