Challenge ‘23 – The Solutions
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2023 – The Egyptians
1. A Pyramid Paint Scheme
Imhotep the architect wants to paint the Great Pyramid. He wants one red side, one blue side, one green side and one yellow side.
How many different ways can he paint it?
His assistant is tasked with making models to show the possibilities. He realises he can make far fewer models as they can be turned around. How many different models does he need to make?
Show how he would need to paint each of the models.
1. A Pyramid Paint Scheme
There are 24 ways to paint the pyramid
Only 6 models are required
2. Don’t Get the Hump
The old pharaoh dies and leaves his herd of 19 camels to his three daughters. Hatshepsut, the eldest, is to receive ½ of the camels; Isis, the middle daughter, is to receive ¼; and Nefertiti, the youngest, is to receive 1/5. Their uncle offers to lend them a camel to help.
How many camels does each
sister receive?
2. Don’t Get the Hump
Hatshepsut | Isis |
Nefertiti | Uncle |
3. Four Flowers Fit
The pharaoh’s gardener needs to plant 16 flowers into a four-by-four grid.
The Pharaoh wants there to be no repeated flowers in a given row or column. The gardener has made a start.
How could the garden be finished?
L | R | P | J |
R |
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P |
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J |
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L = Lotus R = Rose P = Poppy J = Jasmine |
3. Four Flowers Fit
L | R | P | J |
R |
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P |
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J |
|
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L | R | P | J |
R | P | J | L |
P | J | L | R |
J | L | R | P |
L | R | P | J |
R | J | L | P |
P | L | J | R |
J | P | R | L |
L | R | P | J |
R | L | J | P |
P | J | R | L |
J | P | L | R |
L | R | P | J |
R | L | J | P |
P | J | L | R |
J | P | R | L |
4. The Son has got his Crown On
The pharaoh’s outfitters, ‘Cloaks and Crowns’, sell cloaks and crowns in 3 sizes – prince, king and emperor (in increasing size order).
He wants to kit out his three sons for the forthcoming festival of Ra. Ahmose chooses a bigger cloak than Kamose, but a smaller crown than Thutmose. Both Thutmose’s cloak and crown are larger than Kamose’s, but the size of Thutmose’s crown matches the size of Kamose’s cloak. Which sizes did the pharaoh’s sons each get?
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4. The Son has got his Crown On
| Cloak | Crown |
Ahmose | K or E | P or K |
Kamose | P or K | P or K |
Tutmose | K or E | K or E |
Ahmose and Tutmose have larger cloaks than Kamose
Ahmose and Kamose have smaller crowns than Tutmose
Kamose’s cloak is the same size as Tutmose’s crown
King
King
Emperor
Emperor
Prince
Prince
5. The 13 Steps
The priestess at Horus’ temple has to go up a flight of 5 steps each day. The high priest has decreed that she will ascend the steps one or two at a time. She wants to know if she can do this a different way each day for a week.
Is it possible?
How many ways are there of ascending the steps?
The grand staircase has 13 steps. The priest wants to know if he can climb this a different way each day for a year using the same rules as the priestess.
Is it possible?
How many ways are there of ascending the grand staircase?
Explain why your answer is correct.
5. The 13 Steps
Yes, it is possible on the five steps.
There are 8 combinations.
1,1,1,1,1 | 1,1,1,2 |
1,1,2,1 | 1,2,1,1 |
2,1,1,1 | 1,2,2 |
2,1,2 | 2,2,1 |
5. The 13 Steps
Yes, it is possible on the 13 steps.
There are 377 possible combinations.
These are Fibonacci numbers, obtained by adding the two previous answers.
Steps | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
Comb’ns | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 |
5. The 13 Steps
To reach the nth step you either:
Climb one step from the (n-1)th step
Or
Climb two steps from the (n-2)th step
So each time, it’s those two previous results added together.
Steps | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
Comb’ns | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 |
6. Rock Hard Sums
An ancient Egyptian stone tablet is found, as shown to the left. It is correctly deduced that it is a sum, with each of the symbols representing a different digit. Surprisingly, there are 2 ways it could be done.
What are they?
6. Rock Hard Sums
From the 100’s column, B<5 otherwise the answer would be over 1,000.
If B = 1, then C = 2, 3, or 4 from the 100’s column.
The units column tells us that C must be odd, so C = 3
If C = 3, the A must be 4 so we get 11 as the sum of the units column.
| A | A |
B | A | C |
B | A | A |
C | C | B |
| 4 | 4 |
1 | 4 | 3 |
1 | 4 | 4 |
3 | 3 | 1 |
6. Rock Hard Sums
If B = 2, then C = 4, 5, or 6 from the 100’s column.
The units column tells us that C must be even, so C = 4 or 6
If C = 4, the A would be 4 or 9 for the units column.
The symbols are distinct, so not 4, and 9 doesn’t work for the 10’s column.
So C = 6, and then A = 3 or 8.
A = 3 won’t work for the 10’s column, so it must be A = 8
| A | A |
B | A | C |
B | A | A |
C | C | B |
| 8 | 8 |
2 | 8 | 6 |
2 | 8 | 8 |
6 | 6 | 2 |
6. Rock Hard Sums
B= 3 or 4 doesn’t work, but we know there are 2 solutions, so can stop anyway.
| 8 | 8 |
2 | 8 | 6 |
2 | 8 | 8 |
6 | 6 | 2 |
| 4 | 4 |
1 | 4 | 3 |
1 | 4 | 4 |
3 | 3 | 1 |
7. Plagues Palaver
It all began when the river’s water turned to blood.
Later, it rained hail & fire.
The flies were immediately after the lice.
3 days of darkness was the 3rd plague after the terrible boils, and the only insect plague between them was the locusts.
The frogs were 3 plagues before the livestock dying.
Hail & fire was before the locusts and the darkness, but after the boils and the livestock dying.
The death of the firstborn was the final plague.
Place the 10 plagues in the correct order. The marks for this question are all for the reasoning – no mark for just the list!
The list of plagues is given here in alphabetical order.
3 Days of Darkness
Boils
Deaths of the Firstborn
Flies
Frogs
Hail & Fire
Lice
Livestock Dying
Locusts
River Turns to Blood
7. Plagues Palaver
River turns to blood is first.
Deaths of the first born is last.
Lice is immediately before Flies.
3 Days of Darkness
Boils
Deaths of the First Born
Flies
Frogs
Hail & Fire
Lice
Livestock Dying
Locusts
River turns to Blood
7. Plagues Palaver
Boils was 3 plagues before 3 days of darkness.
Locusts are between them, so Lice and Flies are not.
3 Days of Darkness
Boils
Deaths of the First Born
Flies
Frogs
Hail & Fire
Lice
Livestock Dying
Locusts
River turns to Blood
7. Plagues Palaver
Frogs was 3 plagues before the Livestock dying.
3 Days of Darkness
Boils
Deaths of the First Born
Flies
Frogs
Hail & Fire
Lice
Livestock Dying
Locusts
River turns to Blood
7. Plagues Palaver
We now have two blocks of 4 plagues, so Lice and Flies must fit between Frogs and Livestock dying.
3 Days of Darkness
Boils
Deaths of the First Born
Flies
Frogs
Hail & Fire
Lice
Livestock Dying
Locusts
River turns to Blood
7. Plagues Palaver
Hail and Fire is before locusts, but after boils.
3 Days of Darkness
Boils
Deaths of the First Born
Flies
Frogs
Hail & Fire
Lice
Livestock Dying
Locusts
River turns to Blood
7. Plagues Palaver
Hail and Fire is also after the Livestock Dying.
So that means the two blocks go this way around.
3 Days of Darkness
Boils
Deaths of the First Born
Flies
Frogs
Hail & Fire
Lice
Livestock Dying
Locusts
River turns to Blood
7. Plagues Palaver
So this is the final order.
To achieve marks here, evidence had to be given of the process of thinking.
3 Days of Darkness
Boils
Deaths of the First Born
Flies
Frogs
Hail & Fire
Lice
Livestock Dying
Locusts
River turns to Blood
Thank you for watching!