The Answers

Age Old ProblemWhen Alice’s Adventures in Wonderland was first published in 1865, the combined ages of Alice Liddell, to whom Lewis Carroll first told the story, and Carroll himself were 46. Carroll was 20 years older than Alice Liddell. How old was each of them at the time?

Age Old Problem

REAL MADAMWhile tumbling down the rabbit hole, Alice passes various shelves, on one of which is a jar of marmalade. In how many unique ways can the letters of MARMALADE be re-arranged to produce new 9-letter Wonderland words? Like much else in Wonderland, the words created do not necessarily need to appear to make sense

REAL MADAM

MARMALADEThere are 9 letters, which means there are 9! ways to arrange them.However there are 2 M’s and 3 A’s. These can be arranged in 2! and 3! Ways respectively.This gives us the division below:

Finally, the questions asks for new words, so we need to exclude MARMALADE itself, leaving 30,239.

M AA AM

9 !2!×3 !

=3628802×6

=30,240

Eat Me, Drink Me

On reaching the bottom of the rabbit hole, Alice finds a bottle labelled ‘DRINK ME’ and a cake with ‘EAT ME’ written on it in currants. She takes a drink from the bottle, which causes her to shrink. A bite from the cake then makes her grow a lot bigger!

Assume that Alice is 120cm tall to begin with, that the drink shrank her height by 60cm and that the cake increased her height by 100cm.

List all of the possible methods for restoring Alice to her original height, using the minimum possible number of drinks and the minimum possible number of bites of cake to achieve this.

Drink

Eat

Drink

Drink

Eat

Eat

Drink

Drink

Eat Me, Drink Me

15 Ways to do this:

Eat Me, Drink Me

D D D D D D D D D D D D D D D

E E E E E E E E E E E E E E E

D D D D D D D D D D E E E E E

D D D D D D E E E E D D D D E

E E D D D E D D D E D D D E D

E D D E E D D D E D D D E D D

D D E D E E D E D D D E D D D

D E E E D D E D D D E D D D D

However, these 3 would mean that her height is negative

So there are 12 remaining solutions.

Rabbit, Rabbit, RabbitThe White Rabbit gets married in January. At the start of February, he and his wife decide to have children; the first pair, one male and one female, is born exactly one month later. Another such pair follows at the same point in each subsequent month. When each new pair reaches two months old, they produce a similar new pair of their own, and repeat this monthly. How many rabbits will be in their family by Christmas Day?

Rabbit, Rabbit, Rabbit

DecNovOctSepAugJulyJuneMayAprilMarchFebJan

Rabbit, Rabbit, Rabbit

DecNovOctSepAugJuly

Rabbit, Rabbit, Rabbit

DecNovOctSep

Rabbit, Rabbit, Rabbit

DecNov

Rabbit, Rabbit, Rabbit

So that’s 144 pairs of rabbits, which is 288 rabbits!

Pairs of SquaresThe King of Hearts has a square photograph of himself and a square photograph of his wife (which is slightly smaller in size) hanging upon his wall.

The Queen has the same two images, also as square photographs. Being a more narcissistic character, however, the image of herself that hangs upon her wall is much larger than her photograph of the King. Amazingly, the combined area of the King’s two photographs is 1865cm2 and this is also true of the Queen’s two photographs!

Given that all the photographs are a whole number of cm along each edge, what is the edge length of each particular photograph?

Pairs of Squares

a and b are similar, c is much larger than d

By trial and error, we find:

a = 29cm

b = 32cm

c = 43cm

d = 4cm

𝑎2+𝑏2=𝑐2+𝑑2=1865

Pairs of Squares

So, the King’s portraits are:

a = 29cm b = 32cm

Pairs of Squares

And the Queen’s portraits are:

c = 43cm d = 4cm

The × FactorNamed after Lewis Carroll, a Carroll Diagram is used for grouping items depending upon whether or not they have specified properties.

Copy out the empty Carroll Diagram shown on the right, for which the entries are missing.

Grouping here is according to whether or not numbers are squares or cubes.

Complete your copy using the following clues:

The smaller numbers in the Carroll Diagram are all factors of the largest number.

The entries in the boxes are all whole numbers which are greater than 1 and less than 100.

Each box in the Carroll Diagram must contain one or more numbers.

  Square number

Not a square number

Cube number    

Not a cube 

number   

The × FactorThe smaller numbers are all factors of the largest number.

The entries in the boxes are all whole numbers which are greater than 1 and less than 100.

Each box in the Carroll Diagram must contain one or more numbers.

  Square number

Not a square number

Cube number 64  8 

Not a cube 

number 4, 16  2, 32

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

Took a Book (or 4)On a particular bookshelf at home, I have 15 different puzzle books and works of fiction. Five of these books were written by Lewis Carroll. While packing my suitcase for a holiday, I decide to take four books for poolside reading, and I select them at random from the shelf. What is the probability, as a proper fraction in its lowest terms, that exactly two of the books that I take away will be authored by Lewis Carroll?

Took a Book (or 4)

Took a Book (or 4)

Pick 1st Book P(L) = 5/15

P(N) = 10/15

Pick 2nd Book P(L) = 4/14

P(N) = 10/14

Pick 3rd Book P(N) = 10/13

Pick 4th Book P(N) = 9/12 LLNN

Pick 3rd Book P(L) = 4/13 P(N) = 9/13

Pick 4th Book P(N) = 9/12 LNLN

Pick 4th Book P(L) = 4/12 LNNL

Pick 2nd Book P(L) = 5/14 P(N) = 9/14

Pick 3rd Book P(L) = 4/13 P(N) = 9/13

Pick 4th Book P(N) = 9/12 NLLN

Pick 4th Book P(L) = 4/12 NLNL

Pick 3rd Book P(L) = 5/13

Pick 4th Book P(L) = 4/12 NNLL

Took a Book (or 4)

To find the probability along each branch, the denominator is always 15×14×13×12=32,760

The numerators are:

Sequence

LLNN 5 × 4 × 10 × 9 = 1800

LNLN 5 × 10 × 4 × 9 = 1800

LNNL 5 × 10 × 9 × 4 = 1800

NLLN 10 × 5 × 4 × 9 = 1800

NLNL 10 × 5 × 9 × 4 = 1800

NNLL 10 × 9 × 5 × 4 = 1800

Took a Book (or 4)

Adding these together gives us