¿ìè¶ÌÊÓƵ

Cinderladdin, who was exceedingly clever, was walking in the woods one
day when she came across a dirty old jogging-boot under a tree. She picked
it up and one of the spikes pricked her finger. Fortunately her parents
had sensibly had their daughter vaccinated against sleeping sickness, and
it had no effect. She took the boot home. Her seven lovely brothers laughed
at her, of course, like they always did, because it was a Nike and Reeboks
were in fashion that week, but that made her all the more determined to
keep it.

That evening, she got some polish and a rag and started to clean the
boot. (Nobody had told her not to polish trainers). As she rubbed the rag
against the sole there was a puff of smoke and a green flash. A wispy, translucent
figure rose from the boot, to tower menacingly above her.

‘Who are you?’ asked Cinderladdin. The creature’s whiskers twitched.
‘I am the Pussy of the Boot,’ it mewed, ‘and provided you can give me three
proofs of purchase from jumbo boxes of kitticarrot, you are entitled to
three magic wishes. I promise to make them come true, for I am sworn to
obey your every whim.’

Cinders rummaged behind a pile of poisoned porridge and found three
unused boxes of kitticarrot (‘Nine out of ten cats expressed a preference’
and ‘Don’t you know cats aren’t vegetarian?’). She ripped off the proofs
of purchase (just as her brothers had earlier ripped off the boxes of kitticarrot)
and presented them to Pussy.

‘Oh heck,’ it said. ‘You’ve got some.’ ‘Yes! Now, I wish . . . that
I could have as many wishes as I wished,’ said Cinderladdin.

‘I knew some smartypants would do that one day,’ said Pussy in disgust.

‘First I wish to be a . . .’ said Cinderladdin.

‘Wait!’ yelled the Pussy of the Boot. ‘Before I grant your
wish, you must solve a puzzle. Correctly, you understand.’

‘But I only want to be a may . . .’

‘As Puzzy of the Boot I demand that you solve a pussle,’ said the creature,
becoming flustered. ‘In the magic land of Katzwiskas, all cats have either
five or seven tails. The seven-tailed cats eat a lot and belong to the species
Fatticus catticus. The five-tailed ones are very finicky and belong to the
species Fussius pussius. One morning a Katzwiskan fishmonger sees a total
of 53 tails sneaking past his window. He knows that both species always
hold their tails proudly erect when raiding a fishmonger’s. How many cats
are about to invade his shop?’

Cinderladdin thought very hard, and gave her answer. She was right,
of course, because she was exceedingly clever. So the Pussy of the Boot
found himself compelled to make her a mayor.

Well, almost.

Clippety-clop went Cinderladdin’s hooves on the road. ‘I meant m-a-y-o-r,’
she neighed grumpily.

‘Sorry about that,’ said Pussy. ‘These little glitches happen.’

‘I wish to become my original self. After which you can make me Lord
Princess of µþ²¹²µ³ó»å²¹»å.’

‘You’ll have to wait until the magic charges up again,’ said Pussy.
‘These old boots don’t have much juice, you know.’

‘Oh yes they do,’ said Cinderladdin.

‘Oh no they don’t,’ said the Pussy of the Boot. They went on like that
for some time, until they were interrupted by a bell.

‘I’ve started so I’ll finish,’ said Pussy. More bells chimed.

‘Are they chiming ‘turn again Cinderladdin, Lord Mare of Baghdad’?’
she asked hopefully.

‘Sorry, no. But there are three bells, and my musically trained ears
tell me that they are ringing a full peal of changes.’

‘What’s that?’

‘Number the bells 1, 2, 3. A change of bells is a sequence in which
each bell rings just once, say 2-3-1. A full peal consists of all possible
changes, each occurring exactly once – except that the final change repeats
the first, which by convention is 1-2-3. No bell may move more than one
place from one change to the next – so 1-2-3 can be followed by 2-1-3, say,
but not by 3-1-2 because bell 3 moves two places.

‘I will change you back into your normal self if you can tell me in
how many different ways a full peal of three bells can be rung.’

Cinderladdin quickly found an answer. Can you?

Farther along the road they came to a bridge. As they crossed it a deep
voice boomed ‘Who’s that clip-clopping over my bridge?’

‘I am called Cinderladdin, and I am going to ancient Baghdad to become
Lord Princess.’

‘Wrong answer,’ said the voice. A warty, evil-looking man clambered
into view. ‘Aaagh!’ screamed Cinderladdin. ‘It is my wicked Fairy Goduncle!’

‘How can you tell?’

‘Because he’s got his underpants outside his trousers!’

‘I see,’ said Pussy. ‘And, even more significantly, I observe that they
are on inside out.’

‘Since it is you, Cinderladdin my dear, I will let you pass if you give
me . . . that boot,’ said the wicked Goduncle in an ingratiating tone of
voice. ‘It is only a dreadful old Nike and they are out of fashion this
·É±ð±ð°ì.’

‘Nay,’ said Cinderella.

‘I thought you’d been turned back to your normal self,’ protested the
Goduncle in a stage whisper.

‘Nay, not neigh. I am not so foolish that I am unaware that I carry
the magic boot that houses the pussy that obeys the boot’s owner’s every
whim. Keep your filthy paws off my filthy boot, fairy Goduncle.’

‘Curses! In that case, you may pass if you help me get my underpants
on the right way round. Still outside my trousers, you understand.’

‘Take them off, turn them the right way round, and put them on again.’

‘Ah, but the Vizier of the Rats has tied my feet together with an unbreakable
. . . and non-untieable . . . braid of Rapunzel-hair.’

‘That makes it a little harder,’ said Cinders. Can you work out what
she did?

The Fairy Goduncle thanked them profusely and drove off on a huge marshmallow
pulled by three bears. They had gone but thirty leagues more when there
was a tremendous crash and an enormous pumpkin hit the ground just ahead
of them. Cinders gathered the remains in her apron.

‘What on earth are you doing that for?’ asked Pussy.

‘It might come in useful one day,’ said Cinders. ‘Waste not, want not.’

‘Look out!’ yelled the Pussy of the Boot, and they rolled into the nearest
ditch. Another large pumpkin splashed messily nearby. Cinders rushed out
and scraped it up. ‘What’s happening?’

‘Look up,’ said the Pussy of the Boot. Above them seven dwarves flew
past on a magic flying carpet, pushing an endless supply of vegetables over
the side.

‘What does it run on?’

‘Carpet shampoo, I think,’ said Pussy. ‘Unthreaded, of course.’

A large parsnip whizzed past and went splat against a nearby troll.
Cinders scraped that into her apron too. ‘Thank god it wasn’t a beetroot.’
she said.

‘Why not?’

‘I don’t want to be marooned.’

‘The magic charge on my boot has built up again,’ Pussy hinted, ‘But
just for that terrible pun, I will only let you wish that the dwarves run
out of unthreaded carpet shampoo if you can answer another of my riddles.’

‘Oh, very well,’ said Cinders. ‘You really are a most tiresome creature,
considering you are sworn to obey my every whim.’

‘Creative marketing,’ said Pussy. ‘Don’t believe the hype. Have you
noticed that those dwarves have just flown to yonder dark mountain peak
and back?’

‘The one with all the lightning?’

‘That very one. ‘Tis the dwelling of the evil wizard Wexwissel. Doubtless
these are his garden produce with which the dwarves are pelting us. Now,
’tis a distance of a hundred leagues – each way – and the carpet is travelling
at fifty leagues per minute . . .’

‘Four minutes,’ said Cinders.

‘No, no, wait for the question. Suppose that the evil wizard casts a
spell to make a huge storm blow up suddenly, coming towards us from the
mountain at a speed of twenty-seven and three-quarter leagues per minute.
Would it take the dwarves longer to make the journey there and back, or
shorter, or exactly the same time?’

‘Exactly the same,’ said a brown cow that suddenly emerged from a passing
pirate ship. It looked as if it were wearing two pairs of pyjamas, and the
back legs were a foot shorter than the front ones. Its stomach seemed to
be smoking a cigarette. It did a quick soft-shoe shuffle and tripped over
a mushroom.

‘How now,’ said Pussy, predictably. ‘Why say you that?’

‘Why speak you so funny?’ replied the back half of the cow. The front
half told the back half to shut up, and after some disagreement they explained
their combined reasoning. ‘Because the extra speed they gain when coming
back will cancel out the speed they lose when going, right?’

Cinders thought for a few seconds. ‘I’m not so sure that argument works,’
she said.

‘What argument?’ said the back end of the cow. ‘Was I arguing?’

‘Yes, you were,’ said the front end.

‘No I wasn’t’

‘Yes you were.’ Look let’s take the next twenty lines for granted, OK?
Is the cow’s reasoning correct? If not, what did Cinderladdin decide?

The flying carpet coughed, spluttered, and fell from the sky like an
autumn leaf carrying an elephant, to impale itself upon the distant peak.
‘Good . . . it was unthreaded shampoo,’ said Pussy.

‘I wish you’d stop talking such rubbish,’ said Cinders in irritation.
‘We can’t waste any more time on this stupid old cow, we’ve got to get to
µþ²¹²µ³ó»å²¹»å.’

‘But I am a magic cow,’ said the cow. ‘I will help you fulfil your heart’s
desire to become Lord Princess of µþ²¹²µ³ó»å²¹»å.’

‘Prove it,’ sneered Cinders.

‘I will give you . . . a magic bean,’ said the cow.

‘Oh, big fat hairy deal.’

The cow held up a leather bag, tied with a thong. It had trouble grasping
the knot in its hooves to untie it. Suddenly the knot gave way, and a torrent
of beans spilled from the bag, to roll away down the hillside. ‘Oops,’ said
the cow. ‘Oh well, beansy come hasbeansy go. Wait till they come to rest,
and – yes, just as I expected.’

At the bottom of the hill there were loud rumblings, and a number of
tree-like plants sprouted skywards at an enormous rate, to disappear into
the clouds. ‘Instant beanstalks,’ said the cow proudly. ‘Precoated with
my own-brand magic fertiliser. By the way, how many beans did I drop?’

Cinderladdin counted the towering plants. ‘I make it forty,’ she said.

‘An auspicious number,’ said the cow. ‘Now, Cinders my dear. At the
top of one of the beanstalks is a glass goose that lays golden slippers.
All of the others are bewitched.’

‘In what way?’

‘There is a wicked witch living at the top of each of them. I want you
to select the one beanstalk that is begoosed. You must guess, for there
is no way to tell them apart except by climbing them. And I assure you,
you don’t want to do that unless you’re pretty damned certain it’s the begoosed
stalk you’re climbing.’

‘U³¾.’

‘Unless you think you will enjoy the happy-go-lucky life of common
bog-toad, of course.’

‘Oh. Can I ask the Pussy of the Boot?’

‘You can ask. But just now you wished it would shut up, and it did,
and the boot-magic was temporarily discharged, and it hasn’t built up again
yet, so I doubt you will get a reply. However, I am not going to ask you
to guess correctly, for your chance of success is very low. My question
will be accessible to rational analysis.’

‘Oh. Um . . . then I choose this one,’ said Cinders, pointing.

‘W³ó²â?’

‘Because it was the one recommended in Witch magazine.’

‘You can’t resist them, can you?’ said the cow. ‘Very well. If you are
right, all the other thirty-nine beanstalks are bewitched. Even if you are
wrong, which heaven forfend, precisely thirty-eight of the remaining thirty-nine
beanstalks are bewitched. I want you to choose a number between one and
thirty-eight, and I will then cause the clouds to roll back and reveal witches
at the top of that number of beanstalks, all being selected from those that
you have not chosen. For example, if you say ‘seven’, I will select seven
beanstalks, other than your guess, and show you that those particular seven
beanstalks are bewitched. Got that?’

‘Um. Think so.’

‘I will not in any circumstances reveal what is at the top of your chosen
beanstalk – be it witch or goose.’

‘I see.’

‘You may think that such information cannot offer you any advantage,
because I can always find enough bewitched beanstalks to reveal, no matter
what your original choice was. But once I have shown you what is at the
top of a certain number of beanstalks, you will increase your chances of
being right if you change your mind and select a different beanstalk from
your original choice. My question is: what is the smallest number of bewitched
beanstalks that I must reveal, for your change of mind to be advantageous?’

‘Hmmm. Well, if you show me thirty-eight witches, either my original
choice was right or the only remaining beanstalk is begoosed. But it could
be either. Each seems equally likely . . . I must give this some further
³Ù³ó´Ç³Ü²µ³ó³Ù.’

Eventually Cinderladdin found the right answer. After which the cow
climbed (a little laboriously) to the top of the correct beanstalk and brought
back the glass goose. Cinderladdin looked after it and fed it every day
on mashed pumpkin (you see, it did come in useful) and the goose produced
millions of golden slippers. Cinders sold them at royal balls up and down
the country. Then she emigrated to Baghdad, purchased a small princessship
(one of the only two words I know with three consecutive s’s – can you find
another?) and lived happily ever after in a straw cottage rented to her
by a pig.

Until one day a passing frog blew it down and stole her porridge.

* * *

1. Four Fattici cattici and five Fussii pussii. You can get this by
trial and error, but here’s a useful trick. The number 53 leaves remainder
3 on division by 5, so the total number of F. cattici tails is a multiple
of 7 that also leaves this remainder. It therefore ends in either 3 or 8.
But 63 is too big, so there must be 28 tails from the F. cattici. That leaves
25 from the F. pussii.

2. There are two full peals: 1-2-3; 1-3-2; 3-1-2; 3-2-1; 2-3-1; 2-1-3;
1-2-3, and its reverse.

3. Slide the underpants down on the rope. Then pass one leg through
the other, and slide them back onto Fairy Goduncle.

4. The cow is wrong. The journey against the wind takes longer than
the journey with it, so the adverse effect of the wind lasts longer than
the helpful effect does. The situation is not symmetrical and the changes
don’t cancel.

5. Even if the cow shows Cinders only one bewitched beansalk, her chances
improve if she then changes her mind. In fact, the more bewitched beanstalks
she is shown, the better the chances become – provided she changes her mind.
At the initial choice shw has a chance of 1 in 40 of being correct. After
being shown 1 bewitched beanstalk, she has a probability of 1 in 38 of being
right if her first choice was wrong, but a probability of zero if it was
right. So her probability of being right after changing is 39/40 x 1/38
which is 39/38 times as big as her original chance of 1 in 40. Since 39/38
is bigger than 1, her cances have increased.

6. How about ‘Sss’ (noise made by a snake)?