### A FAMILY QUESTION

```The Answers
Black Magic
Egrun, the wild green witch of the Wirral, roars along its lanes
at nightfall astride her wood-burning Kawasaki broomstick.
One night, she rides from the former Cadbury’s factory in
Moreton at midnight to visit the Wizard of Oswestry for a short
spell. But when she gets to Chester, her faithful catnav,
Subordinate Claws, reminds her that she has forgotten the
chocolate. So she goes all the way back for it. It’s 20 miles
from Moreton to Chester and 30 miles
from Chester to Oswestry. Egrun always
rides at a steady 120mph and nothing
EVER gets in her way. At what time
does she arrive in Oswestry?
Black Magic
Moreton
00:45
00:20
00:10
Chester
Oswestry
Pick and Mix
Robert and Miriam need chocolates as prizes for a quiz:
they decide on four bags of Revels, five Mars Bars and five
Twix Bars. However, they don’t coordinate their shopping!
Robert buys three bags of Revels, two Mars Bars and four
Twix Bars for £5.85;
Miriam separately purchases two bags of Revels, four Mars
Bars and three Twix Bars for £5.60.
They return the excess goodies to the
shop, receiving £2.60 back.
How much does each item cost?
Pick and Mix
This is best considered as three simultaneous
equations:
3r + 2m + 4t = 585
[1]
2r + 4m + 3t = 560
[2]
r + m + 2t = 260
[3]
2r + 2m +4t = 520
[4] = [3] × 2
r = 65
[5] = [1] – [4]
Pick and Mix
Sub into [3] → m + 2t = 195
[6]
sub into [2] → 4m + 3t = 430 [7]
4m + 8t =780
[8] = [6] × 4
5t = 350, so t = 70
[8] – [7]
Sub r = 65 and t = 70 into [3] → m = 55
So Revels cost 65p, Twix cost 70p, Mars cost 55p
Who’ll Nut?
Fred and George are to share a chocolate bar made up of an 8
by 6-square rectangular array. The top-left & top-right squares
each contain a visible nut. They take it in turns, starting with
Fred, to snap the chocolate into 2 pieces along a line between 2
rows or 2 columns, & then eat 1 of the 2 pieces. Neither Fred
nor George wants a nut. Which person can guarantee that they
won’t get the nut? Is this true for any size of bar?
Who’ll Nut?
Ingrid, Jean, Karen, Philippa and Shirley are sharing a box of
chocolates.
There are 5 chocolates left in the box:
a Coffee Cream, an Orange Fondant, a Hazelnut Praline, a
Toffee Crunch & a Nougat Supreme.
They agree to have one chocolate each. Ingrid likes only
Orange Fondants and Toffee Crunches; Philippa likes only
Orange Fondants and Hazelnut Pralines; Jean and Shirley both
like Coffee Creams, Hazelnut Pralines and Nougat Supremes;
and Karen likes only Toffee Crunches
and Nougat Supremes.
List all the possible ways that the
chocolates can be shared such that
all of the ladies get one that they like?
Ingrid
Coffee Cream
Jean
Orange Fondant
Karen
Hazelnut Praline
Philippa
Shirley
Toffee Crunch
Nougat Supreme
The Trouble With Truffles
There are 9 chocolates in a selection box: 3 white, 3 milk & 3 dark
chocolates. There are 3 soft centres, 3 truffles & 3 pralines. Each
chocolate is different. Using the clues below, arrange the
chocolates in a 3 x 3 grid:
The 3 truffles are in the central column;
there is 1 of each colour in each column;
there are no 2 soft centres are next to each other horizontally,
vertically or diagonally; the same applies for the pralines;
the white praline is in the bottom-right corner; the 3 whites are
arranged along a diagonal line;
there are no dark chocolates in the
bottom row.
The Trouble With Truffles
The 3 truffles are in the central column;
there is 1 of each colour in each
column;
there are no 2 soft centres are next to
each other horizontally, vertically or
diagonally; the same applies for the
pralines;
the white praline is in the bottom-right
corner;
the 3 whites are arranged along a
diagonal line;
there are no dark chocolates in the
bottom row.
Be Twix and Between
How many different ways can you arrange the letters of the
word CHOCOLATE?
There are 9 choices for the first letter, 8 for the
second and so on.
This gives us
9×8×7×6×5×4×3×2×1 = 362,880 = 9!
However there are 2 ‘C’s and 2 ‘O’s, so we need
to halve this number for each of them
362,880 ÷ 2 ÷ 2 = 90,720
Chocolate Brownies
Seven Brownies and three Leaders have baked 35
chocolate cupcakes and eaten them all.
cupcakes, and the third Leader ate three cupcakes.
Only whole cupcakes may be eaten.
Show that at least one (greedy) Brownie ate at least 5
cupcakes.
Chocolate Brownies
Brown Owl
Barn Owl
Tawny Owl
Brownie 1
Brownie 2
Brownie 3
Brownie 4
Brownie 5
Brownie 6
Brownie 7
13
20
27
34
35
1
3
6
Cupcakes
Cupcake
Fry’s Delight
Miss Fry was given a box containing a
variety of chocolates. Although she likes
chocolates, she is not greedy, so she
decided to share her chocolates and make
them last. Her method of consumption was
to eat one on the first day, and give away
10% of the remainder, to eat 2 on the
second day and give 10% of the remainder
away, eat three on the third day and give
away 10% of the remainder, and continue
in this way until no chocolates were left.
How many chocolates were in the box and
how many days did they last?
Fry’s Delight
We begin with the assumption that she
doesn’t eat or give away parts of chocolates.
This means that the minimum she can give
away is 1.
1 is 10% of 10, which means she would
have 9 left to eat, which would make it the
9th day.
Working backwards gives:
Fry’s Delight
Day
9
8
7
6
5
4
3
2
1
No at
start
9
18
27
36
45
54
63
72
81
Eat
Leaving
9
8
7
6
5
4
3
2
1
0
10
20
30
40
50
60
70
80
Give
Away
0
1
2
3
4
5
6
7
8
End of
Day
0
9
18
27
36
45
54
63
72
Bournville Dreams
A chocolate factory makes two different types of
chocolate: dark and light. Each day, it receives
1200L of milk and 1200kg of cocoa. A batch of
dark chocolate requires 20L milk and 50kg of
cocoa. A batch of light chocolate requires 40L of
milk and 30kg cocoa. The management insists that
the workers produce at least 25 batches in total
each day and that only full batches can be
produced. The company makes a profit of £15 on
a batch of dark chocolate and £10 on a batch of
light chocolate. Work out how many batches of
each type need to be produced to maximise
profits.
Bournville Dreams
A chocolate factory makes two different
types of chocolate: dark and light. Each
day, it receives 1200L of milk and 1200kg
of cocoa. A batch of dark chocolate
requires 20L milk and 50kg of cocoa. A
batch of light chocolate requires 40L of
milk and 30kg cocoa. The management
insists that the workers produce at least
25 batches in total each day and that only
full batches can be produced. The
company makes a profit of £15 on a batch
of dark chocolate and £10 on a batch of
light chocolate. Work out how many
batches of each type need to be
produced to maximise profits.
First we identify
constraints.
Cocoa
50d + 30l ≤ 1200
Milk
20d + 40l ≤ 1200
Min Batches
d + l ≥ 25
d ≥ 0, l ≥ 0
Bournville Dreams
Bournville Dreams
Number of Light Batches
Maximise: profit = 15d + 10l
The optimum solution will always be at a corner of
the feasible region. These are:
26
• (0,30) = £300
25.9
• (0,25) = £250 25.8
25.7
• (22,3) = £360
25.6
25.5
• (9,25) = £385
25.4
• (8,26) = £380 25.3
25.2
• (8,25) = £370
25.1
25
Milk = 9 × 20 + 25 × 40 = 1180L
Cocoa = 9 × 50 + 25 × 30 = 1200kg
8
8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9
Number of Dark Batches
9
Christmas Celebrations
At Christmas, Ken likes to buy chocolates for
family, friends and carol singers. He decides to
spend £200 and has 100 people to buy for. He
divides his spending between family-sized tins for
£10, small selection packs for friends at £2, and
fun-sized chocolate bars for carol singers at 20p
each. Given that Ken has more friends than he
anticipates carol singers, how many of each item
Christmas Celebrations
T + S + B = 100
10T + 2S + 0.2B = 200
B must be a multiple of 5 and S > B
2T + 2S + 2B = 200
8T - 1.8B = 0
8T = 1.8B
40T = 9B
Christmas Celebrations
40T = 9B
B = 0, 40 or 80
 S > B, so B ≠ 80
B = 0 => T = 0 => S = 100
B = 40 => T = 9 => S = 51
So 40 carol singers, 9 family and 51 friends.
Box Clever
chocolate powder is to be made from a
single square sheet of card, with side
length of 24cm, by cutting smaller
squares (of integer side length) from
the corners of the original square. It will
then be folded up to make an open box
What is the maximum volume that can
be obtained in this way? Show, using a
graph, that this is the maximum
volume.
Box Clever
cut out
0
1
2
3
4
5
6
7
8
9
10
11
12
base
24
22
20
18
16
14
12
10
8
6
4
2
0
volume
0
484
800
972
1024
980
864
700
512
324
160
44
0
Box Clever
1200
Volume of box (cm³)
1000
cut out
0
1
2
3
4
5
6
7
8
9
10
11
12
800
base
24
22
20
18
16
14
12
10
8
6
4
2
0
Maximum
volume of
1024cm³ when
cut out is 4cm
600
400
200
volume
0
484
800
972
1024
980
864
700
512
324
160
44
0
0
0
1
2
3
4
5
6
7
8
Size of cut out (cm)
9
10
11
12
Six After Eight
A small selection bag of posh
chocolates. These are a random
selection from 8 different
chocolates. A quality-control system
is put in place to ensure that no
selection contains more than 2 of
the same chocolate or fewer than 4
different chocolates. How many
different bags of chocolates are
possible?
Six After Eight
There are 3 possible scenarios:
6 unique chocolates
1 pair and 4 other unique chocolates
2 pairs and 2 other unique chocolates
Six After Eight
6 unique chocolates
8!
6! 2!

8 7
2
 28
Six After Eight
1 pair and 4 other unique chocolates
8
7!
4! 3!

8 7 65
3 2
 8  7  5  280
Six After Eight
2 pairs and 2 other unique chocolates
8 7
2

6!
2! 4!

8 7 65
4
 12  7  5  420
Six After Eight
28 + 280 + 420 = 728
Seg-sational!
There are 20 segments in a
chocolate orange. Modelling a
chocolate orange as a sphere of
diameter 5cm, with a cylindrical
hollow of diameter 5mm running
down the core, what is the
volume of a segment?
You may discount the small
dome of chocolate at each of the
‘poles’ of the Chocolate Orange.
Seg-sational!
Volume of sphere of chocolate
4
3
r 
3
4
3
  2 . 5  65 . 45 cm
3
3
Seg-sational!
Volume of hollow cylinder
 r h    0 . 25  5  0 . 98 cm
2
2
3
Seg-sational!
Volume of segment
65 . 45  0 . 98  64 . 47
64 . 47  20  3 . 22 cm
3
Breaking the Mould
Anna and Billy play a game with a
rectangular chocolate bar that is 5
squares by 10 squares. Anna starts. They
take turns breaking a piece of the bar
(only one piece can be broken in one
turn, always along the lines between the
squares). The first player to break off a 1
by 1 square piece wins. Who wins the
game?!
Breaking the Mould
Anna
Billy
A Year’s Supply of Chocolate
Assume that all of the Cadbury’s Dairy
Milk bars bought in England were the
standard length of 11.4cm. When laid
end-to-end, they would form a line around
the Earth along the line of latitude
passing through Liverpool (53½°N).
Using the Earth’s circumference at
40,075km at the equator, how many bars
are sold in England each year?
A Year’s Supply of Chocolate
40075 ÷ 2π = 6378.134km
6378.134 × cos (53.5) =
3793.860km
Circumference at Liverpool
= 3793.860 × 2π
= 23837.523km
A Year’s Supply of Chocolate
Circumference at Liverpool
= 3793.860 × 2π
= 23837.523km
= 2383752317.9cm
Number of bars
= 2383752317.9 ÷ 11.4
= 209101080.5118
≈ 209,101,081 bars
```