# Summer Challenge

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```Challenge 1
Arrange all the digits 1 to 9, using
them once each only, to form a
fraction equivalent to 1/3
Challenge 1 Hint
What could the 2 numbers end in?
What could the first digit of the
bottom number be?
Challenge 2
A handywoman is going to fix numbers onto
bedroom doors in a very large hotel. The
numbers are supplied as individual digits.
There are 1000 rooms, how many 7’s does she
need?
Challenge 3
Why is the number 8,549,176,320 special?
Challenge 4
What are the smallest integer values of A, B and
C such that the product of any two of the values
added to the third gives a square number?
Challenge 5
Find the three 4 digit numbers that equal the
square of the sum of the two 2-digit numbers
formed from the 1st and 2nd digits and 3rd and
4th digits.
e.g. one that doesn’t work:
1546 ≠ (15+46)2
Challenge 5 Hint
Can you use a spreadsheet to help you find
them?
Challenge 6
.-- .... .- - -.. --- -.-- --- ..--. . - .. ..-. -.-- --- ..-- ..- .-.. - .. .--. .-.. -.-... .. -..- -... -.-- -. .. -. .
Help here
Challenge 7
A set of 15 dominoes (all combinations of 1 to 5)
is placed on a square grid as shown – except that
the outlines haven’t
2 2 3 5 4 3
been given…
What number is on the
2 1 3 2 5 4
other half of the domino 4 1 5 5 3 4
with the ‘1’ highlighted? 3 3 2 2 4 1
4 5 1 5 1 1
Challenge 7 Hints
What are all the possible dominoes?
Are there any dominoes
that have to be in a
certain place?
2
2
4
3
4
2
1
1
3
5
3
3
5
2
1
5
2
5
2
5
4
5
3
4
1
3
4
4
1
1
Challenge 8
Cherries are naturally 80% water.
Dried cherries are made by leaving them in the
sun until they have lost 75% of their water.
What is the percentage water content
of dried cherries?
Challenge 9
Mia uses an escalator at a railway station.
If she runs up 8 steps of the escalator, then it
takes her 55 seconds to reach the top. If she runs
up 15 steps of the escalator, it takes her 37.5
seconds to reach the top.
How many seconds would it take
Mia to reach the top if she did not
run up any steps at all?
Challenge 10
Each side of the triangle ABC is divided into 5
equal parts.
What is the ratio of the area of the blue triangle to
the area of ABC?
B
A
C
Challenge 10 Hints
B
C
A
A
B
C
Challenge 11
A boat carries rocks on a small, still lake making
the depth of water in the lake D.
The rocks are released and sink to the bottom of
the lake.
Is the level of water in the lake now greater than,
less than or the same as D?
Challenge 11 Hint
Think about a small marble made of gargantuan –
a (fictitious) metal which is 1000 times as ‘heavy’
as lead.
Challenge 12
A large number of sweets need to be eaten.
Eating alone, it takes Hannah an hour to eat a jar
of them, Mike 3 hours, Nat 5 hours and Oscar 7
hours.
Eating together, how long, to the nearest
minute, does it take them to eat their
way through 3 jars of sweets?
Challenge 12 Hints
In 15 hours, Mike will have eaten 5 jars and Nat
will have eaten 3 jars.
Can you find a whole number of hours in which
they will all have eaten full amounts of jars?
Challenge 13
A symmetrical yellow crescent is formed from two
circles as shown with O being the centre of the
larger circle.
AB = 5cm and CD = 9cm
What are the diameters of
the two circles?
Challenge 14
The circumference of a circle is divided into n
equal arcs and semi-circles constructed as
shown.
Can the blue shapes at the edge ever be equal in
area to the orange petals?
If so, what should n be?
Challenge 14 Hint
Try looking at just one blue shape and orange
petal.
Challenge 15
What is the internal side length of the smallest
hollow cube which can wholly contain 4 identical
spherical chocolates, each of diameter 10cm?
Challenge 15 Hint
Think about how the chocolates could be
arranged. Some ping pong balls or tennis balls
might be helpful.
Teacher notes: Summer Challenge
The final edition of the academic year is a replica of the delegate
challenge from MEI’s Annual Conference.
There are 15 challenges, presented in approximate order of difficulty,
for you and your students to enjoy. You should find some questions
are suitable for KS3 students whilst others will challenge A level
students.
Some hints are provided after the relevant question and all answers are
given in the teacher notes below.
Enjoy!
Teacher notes: Summer Challenge
Challenge 1
5832
17496
or
5823
17469
Challenge 2 300
Challenge 3 All the digits are in alphabetical order
Challenge 4 1, 7 and 9
Teacher notes: Summer Challenge
Challenge 5 2025 3025 9801
Sample spreadsheet columns below, searching for zero values in
column G.
For this solution it helps to work ‘backwards’ through the problem.
Start with a number (A) and square it (B), split the square numbers into
first pair and second pair (C leading to D&amp;E), add them together (F),
compare with original number (G).
A
B
C
D
E
F
G
44
1936
19.36
19
36
55
11
45
2025
20.25
20
25
45
0
n
n^2
B/100
int( C )
(C-D)*100
D+E
F-A
Teacher notes: Summer Challenge
Challenge 6 What do you get if you multiply six by nine? 54
Challenge 7 4.
Challenge 8 50%
Challenge 9 75 seconds
Challenge 10 8:25
look for triangles of equal base and height.
Challenge 11 Less than D.
When the rocks are in the boat, the mass of water displaced is equal to
the mass of the rocks, when in the water, the volume of water displaced
is equal to the volume of the rocks.
Since the rocks sink, their density is greater than 1, hence more water
is displaced when the rocks are in the boat than when they are in the
water.
Teacher notes: Summer Challenge
Challenge 12 1 hour and 47 minutes
Find the lowest common multiple of the number of hours for each
person to eat a jar.
In 105 hours, Hannah eats 105 jars, Mike 35, Nat 21 and Oscar 15.
That’s 176 jars between them.
176 jars in 105 hours (6300 minutes), so use proportional reasoning to
find how long it takes to eat 3 jars.
Challenge 13 50cm and 41 cm
Teacher notes: Summer Challenge
Challenge 14 8
Consider the circle to have a radius of 2r, then the
area of the circle is 4πr2
Looking at a single sector and ‘petal’,
blue area=orange area
only if
area of sector = area of semicircle
Area of semicircle is 0.5πr2
Area of sector = 0.5πr2
when there are 8 sectors
Teacher notes: Summer Challenge
Challenge 15 10+5√2 or an equivalent expression
The chocolates should be arranged as a tetrahedron with each
chocolate nestling in a vertex. This means that 2 chocolates will sit
diagonally at the bottom of the box and 2 at the top. Use this 2d
diagram to calculate the size of the square.
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