Senior Challenge ‘23�The Solutions
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2023 – The Egyptians
1. Seti’s Party
Seti held a party. He says, ‘There were 11 people at my party, and everyone had 3 friends there.’
Is this possible? Explain your answer.
1. Seti’s Party
If each of the 11 people has 3 friends, that makes 33.
However, each friendship is reciprocal.
If John is friends with Amy, then Amy is friends with John.
This means each friendship is counted twice, so 33 would need to be 2 × number of pairs, which is impossible.
Common misconception that Seti wasn’t included in the 11.
2. Egyptian Fractions
The ancient Egyptians used only unit fractions, so they would write ¾ as ½ + ¼.
Find the six ways to write 1 as the sum of 4 distinct unit fractions.
Write 1 as a sum of 5 distinct unit fractions, without using ½.
2. Egyptian Fractions
2. Egyptian Fractions
3. A Horse With No Name
Father and son make a 64km journey through the desert, starting at 4pm. They only have one horse (with no name), which can travel at a steady 8km/hour carrying one of them.
Father can maintain a steady walking pace of 3km/hour and son a pace of 4km/hour. They alternately ride and walk. Each one ties the horse up after riding a certain distance, then walks ahead leaving the horse for the other’s arrival.
At the half-way mark, they meet and stop for a
three-hour nap and to feed the horse.
After the nap, they resume the same pattern.
What time do they arrive at their destination?
3. A Horse With No Name
3. A Horse With No Name
3. A Horse With No Name
1600
12km
32km
1730
2000
2230
Nap time!
3. A Horse With No Name
0130
44km
64km
0300
0530
0800
Arrived!
4. Time is Running Out
Cleopatra and her brother, Ptolemy, each have cylindrical water clocks (so they drain uniformly), but their clocks have different heights and diameters.
Cleopatra’s clock drains from full down to empty in 4 hours and Ptolemy’s does so in 5 hours. After two hours of draining, the water height in both clocks is the same.
What fraction of the height of Cleopatra’s clock is the height of Ptolemy’s?
After each has been draining for three hours, what fraction of the water height of Ptolemy’s clock is equal to the water height of Cleopatra’s?
4. Time is Running Out
4 hours to drain
5 hours to
drain
Cleopatra Ptolemy
4. Time is Running Out
Cleopatra Ptolemy
4. Time is Running Out
Cleopatra Ptolemy
5. Getting to the Heart of the Matter
Somehow, Anubis has managed to lose the feather of Maat! (I guess even gods aren't perfect!)
In the important funerary ceremony, a person’s heart is balanced against this feather. If it is lighter, it means that they are without sin (pure of heart) and are entitled to join Osiris in the field of reeds for their afterlife. Otherwise, they are eaten by the part-crocodile-part-lion-part-hippo
god Ammut and just disappear.
8 people have just arrived to be judged. Fortunately, it is known that only one of them is pure of heart, whilst the remaining 7 are sinners, and have heavier hearts (but all sinners’ hearts weigh the same as each other).
Help Anubis identify the lightest of the 8 hearts, using just the hearts and a balance. But beware! Ammut grows impatient: you may only use the balance twice!
5. Getting to the Heart of the Matter
Put 3 hearts on each side:
If it balances, the sinner is in the unused group
If it tips, the sinner is in the heavier group
5. Getting to the Heart of the Matter
Either way, you now have 3 hearts left.
Put one on each side:
If it balances, the sinner is the unused heart
If it tips, the sinner is the heavier heart
6. Reed the Question
At a temple to Sekhmet, there is a circular reed bed to be planted with 4 different types of reed, one in each of the four sections, as shown here. The radius is 360cm and there are two strings crossing at right angles of lengths 560cm and 640cm.
Find out how far from the centre of the circle the crossing point is.
6. Reed the Question
AF is 560cm
CG is 640cm
OB and OD are perpendicular to the two strings, so that B and D are the midpoints of the strings.
Thus AB is half the length of AF, so 280 cm,
and CD is half the length of CG, so 320 cm
We want the length OE
R = 360cm
A
B
E
D
C
O
G
F
6. Reed the Question
R = 360cm
A
B
E
D
C
O
G
F
6. Reed the Question
R = 360cm
A
B
E
D
C
O
G
F
7. The Unfinished Pyramid
7. The Unfinished Pyramid
Thank you for watching!