“HOLIDAYS are great,” said Millie O’Nair, as she rummaged through the last
two of the six suitcases in search of space to pack her best earrings, “except
for the packing.” She grimaced, levered one of the cases closed, and attacked
the combination locks on the remaining one.
“Let me help,” said her teenage daughter Deb. “I love packing.”
“OK, Debs. You go and help Dad pack this lot in the boot.” Billy O’Nair had
bought the family car —a Redundant Redstart—at an auction, declaring
it to be “a robust machine of total reliability”. Millie agreed that this
sentence was entirely accurate— except for the syllables “ro” and
“r±đ”.
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There was a lot of shouting coming from the front driveway. Deb and Billy
arguing about how to fit the cases into the boot, no doubt. Millie sighed. No
option but to go and sort it out for them, as usual.
All six cases were heaped up on the lawn. Billy had opened the Redstart’s
boot, and removed the rear parcel shelf so that the cases could be stacked to
the roof. Millie watched while they got five cases in place—but the sixth
wouldn’t fit. Then they began shouting at each other again.
“All right, you two, what’s the problem?”
“Dad keeps putting them in wrong!”
Billy glared at her. “No I don’t, Debs. There’s something funny about the
sizes—it looks as if they ought to fit easily, but they don’t seem to. All
of the cases are the same— rectangular blocks two feet square and one foot
łŮłóľ±ł¦°ě.”
“And the boot?”
“With the shelf removed there’s an exact 3-foot cube available.”
Millie thought for a moment. “So the capacity of the boot is 27 cubic feet,
and there are six cases, each holding four cubic feet, a total of 24. Which
means that there ought to be three cubic feet to spare.”
“Right. But there isn’t. It’s not just the size, but the shape that
łľ˛ąłŮłŮ±đ°ů˛ő.”
Millie pursed her lips in contemplation. “Must be some way to—aha!”
“Aha what?”
“You’re going about this all wrong. The reason you can’t solve the problem is
that both of you are worrying about packing the cases into the boot.”
Billy stared at her. “But what else should we be trying to pack in? There
isn’t anything else!”
Millie gave them a Mona Lisa smile. “Oh, but there is.”
…
It was nearly midnight when the car pulled into the entrance to the Shades
Hotel. At least, that’s what Billy presumed it called itself: the initial S must
have fallen off.
The landlady was there to greet them. She introduced herself as Mrs Grimm.
“Payment in advance,” she told Billy.
It was too late to argue, they all needed a rest. Deb was fractious and
Millie was exhausted. “Let’s see… we’re booked in for five nights,
at—how much per night?”
“The price is a whole number of pounds,” said Mrs Grimm, “which exceeds its
reversal by one-eleventh of the digits’ total.” She waved her finger at Deb, who
was about to speak. “And the answer is not zero, Miss Clever-clogs. It’s one of
the little tests I set my guests to keep them on their toes. You’ve got three
minutes to solve it, otherwise the price goes up to £100 a night.”
Billy groaned.
…
When they came down to breakfast next morning, the main table, an enormous
circular one with six places laid out, was occupied by three couples—Mr
and Mrs Abbot, Mr and Mrs Bishop, and Mr and Mrs Cardinal. The O’Nairs squeezed
round a tiny three-seater table to one side.
The waitress came in and said something to the six guests. Suddenly everybody
at the main table was getting up and moving around. Twenty minutes later they
were still shuffling about from chair to chair, and nobody had come to take the
O’Nairs’ order. The waitress was looking harassed, and the toast had long ago
gone limp and cold. Finally Billy could stand it no longer. “Excuse me, young
lady, but what is this? The Mad Hatter’s tea party?”
She turned round, embarrassed. “I’m very sorry, sir, but Mrs Grimm has very
strict rules about seating. You see, no married couple may sit next to each
other, no gentleman may sit next to another gentleman, and no lady may sit next
to a lady.”
“Oh, for heaven’s sake!” exclaimed Billy. “Surely you’ve got enough gumption
łŮ´Ç…â¶Äť
“Yes, but there’s another rule, at breakfast only. A husband and wife may not
sit directly opposite each other across the table.”
“Why not?” said Billy.
“Have you looked at yourself in the mirror this morning, sir?”
“Oh. Right. But surely it can’t be that hard to sort out some arrangement so
that Mrs Grimm is satisfied and we can all get served.”
…
On the last day of the holiday the sun came out. At that point Millie
discovered that she had forgotten to pack her swimsuit, so she had to go
shopping for one while the others lazed on the beach.
Eventually, she found the only swimwear shop in the town: Bikini Fallout. In
the window was an attractive blue swimsuit, just her size. Things were looking
up. She went in and found the sales assistant lurking behind a display of beach
towels.
“Swimsuits?” She shook her head. “Sorry, madam, but we don’t stock swimsuits.
There’s not much call for them.”
“But this is a seaside resort. And you’re a swimwear shop.”
“Oh, yeah. Um… well, the weather’s bad.”
“Not today, it isn’t. Anyway, you’ve got one in the window.”
The assistant looked blank for a moment. “Oh. That swimsuit. No, you can’t
have that one.”
“Why ever not?”
The assistant burst into tears. “Look, I’m sorry, but I had an awful day
yesterday, I can’t balance the books, I’ve lost the stocktaking records, and Mr
Savage-Brut is going to be terribly angry when he comes in
tomorrow. That swimsuit is part of the window display, which Mr Savage-Brut
put together himself. He’s extremely proud of it, and he’d blow his top if we so
much as touched it—even to sell something.”
Millie’s heart went out to the poor woman. “Then I’ll buy another one, it
doesn’t matter.”
“Sorry, but—well, you see, we’ve only got the one.”
Millie found this hard to believe.
“Well, the truth is, madam, that yesterday we did have rather a lot. But then
10 coachloads of tourists arrived, one by one. The first coachload bought half
of the stock of swimsuits plus half a swimsuit, and then—”
“Half a swimsuit? Were they French tourists?”
The assistant looked baffled. “No, they came from Barnsley. Anyway, then the
second coachload bought half of what was left plus half a swimsuit, and so on
until all 10 coachloads had finished. Then we had just this one left. No new
stock until next week.”
“Oh dear. It’s the last day of my holiday, and the first one with any
˛őłÜ˛Ô˛őłóľ±˛Ô±đ.”
The sales assistant came to a decision. “Tell you what, madam. As I said,
I’ve made a bit of a cock-up in the stocktaking department and I don’t actually
know how many swimsuits there were to begin with. Mr Savage-Brut’s going to be a
lot more angry about that than about his precious window display. If you can do
me a favour by working out how many swimsuits we sold to the coach parties, then
you can have the one in the window and I’ll think up some excuse to explain why
it’s gone.”
…
“Soon be home now,” said Billy, turning off the M99 on to a tiny country
lane. The Redstart’s motor coughed, faltered, and then picked up again. Billy
breathed a sigh of relief.
“Is this the right road?” asked Millie.
“There are road works at the A777,” said Billy. “This is a short cut.”
“Doesn’t it cross the railway line?” asked Deb.
“Twice,” said Billy. “There’s a loop. But it only takes a few minutes for a
train to… whoops!”
The car screeched to a halt. The level crossing was completely blocked. Two
trains were gridlocked across the loop. They had entered from different
directions at the same time, and each was too long to fit in the loop in the
track.
The two drivers were standing by the road, arguing. Billy managed to calm
them down. “I don’t need to ask what the difficulty is,” he said, “but how did
you get into such a mess in the first place?”
“It’s not my fault,” said the driver of the blue train. “The line to the west
of this crossing is owned by SnailTrack and the line to the east is owned by
RailTrick. The scheduling for SnailTrack is done by Westex Timetables, but that
for RailTrick is done by Eastern Telecom Rotas. Unfortunately there seems to
have been a lack of communication, probably a foul-up at InterRail Liaison
±č±ôł¦.”
“So why don’t you just back off and let me through, eh? I’m very tired and
I’ve only got another couple of miles till I get home. You’re obstructing the
°ů´Ç˛ą»ĺ.”
The driver of the red train shrugged his shoulders. “Sorry, guv, but Southern
Barriers have standing regulations. Once the barrier goes down, it can’t go up
again until the train’s gone past.”
“Yes, and Northern Crossings have a similar regulation,” said the other
driver.
Millie climbed out of the car and her husband explained the difficulty. “So
call in a few more engines,” she said, “and they’ll soon help you sort it
´ÇłÜłŮ.”
The two drivers stared at each other. “Has EasTransCo got any spare
Locomotive Power Units?” asked one.
“Not since last April when we sold them to a bunch of Chechen
entrepreneurs… WesTransCo?”
“Nope, nary a one. Inefficient to keep unused spares, the CEO said at the
last shareholders’ meeting in Barbados.”
“I see,” said Billy. “Then we’ll have to manage with what we’ve got.” He
counted the carriages. “You’ve each got one engine and sixteen carriages. Each
branch of the loop can hold up to eight carriages, or seven carriages and one
engine, right?”
â€Áčľ±˛µłółŮ.”
“Fine. Then you’d better get on with it.”
â€ÂŮ´Ç°ů°ů˛â?”
“Get these trains past each other, correctly assembled with the engines in
front. And do it with as few changes in the direction of engines’ movement as
possible.” The driver gave him an amused look. “But that’s RailShunt’s
responsibility, not ours.”
Billy took out his diary, and consulted it. “Today is the third Tuesday of
the month, which means that under the Level Crossing Users’ Charter I am
entitled to compensation of ÂŁ15.75 for every full minute you keep me
waiting beyond half an hour. I imagine it will be docked from your wages. And
we’ve already been stuck here for 20 minutes.”
The drivers looked at him, then at each other, and rushed off to their
respective trains.
“I didn’t know there was a Level Crossing Users’ Charter,” said Millie,
impressed.
“Didn’t you? Well there is, though what it actually offers is a 10p lollipop
voucher for every week you’re kept waiting.”
…
The Redstart, Millie told her husband, should really be called the Nonstart.
But finally that nice man had stopped and lent them his jump leads, so it had
only taken two hours to get home once the trains had gone.
Millie had no trouble getting Debs straight off to bed. Billy staggered in
with a pile of suitcases, two of which burst, spilling dirty clothes all over
the floor. He sagged into a chair while Millie poured out two stiff gins. Billy
knocked them both back, then saw the look in his wife’s eye. “Sorry, thought
they were both for me. Here, let me get…”
The doorbell rang.
It was Mrs Bung from next door. “Just dropped by to see how you are,” she
said. “Oh, the council left a note to say that they’ll be digging up the road
outside starting at 4 o’clock tomorrow morning. And then they’ll be cutting the
electric off just before breakfast.”
She noticed the horrified expression on Billy’s face. “Never mind, Mr O’Nair.
I’m sure you’ll be lovely and refreshed after that relaxing holiday you’ve had.
And the road works should only last for six weeks, so they say.” She turned
towards the front door. “Nice to get away from it all, isn’t it?”
* * *
Answers to Puzzles
-
Millie is absolutely right. The key is to work out first of all where the
holes should go. There must be three of them, each a one-foot cube. Imagine
dividing the space in the boot into three layers, each of nine one-foot cubes.
Whichever way you put a suitcase in, it must occupy an even number of cubes in
each layer. So as each of the three layers starts with an odd number of cubes,
you have to make sure that the suitcases leave a hole in each one.
Exactly the same argument holds good if the boot is divided up from side to
side or from front to back. This implies that the holes must run along a long
diagonal of the cube—for instance, bottom front left, centre, and top back
right. Once the holes are placed correctly, it is easy to fit the suitcases
around them. -
The price is a whole number greater than 0 and less than 100, so it has
two digits; the second one may be zero. Call them a and b.
Then the price is “ab” (which represents 10a + b) and
the reversal is ba (10b + a). The difference between
them is 9a– 9b. The total is 11a + 11b
, and one-eleventh of this is a + b, so 9(a –
b) = a + b. Therefore a + b is a multiple of
9. If a = b = 9, then 9(a – b) = 0,
which won’t work. So a + b = 9 and therefore a
– b = 1. It follows thata = 5, b = 4, so the
price is ÂŁ54 per night. -
Let the ladies be A, B, C and their husbands a, b, c. Consider A. She must
be seated between two men, neither of them a. So part of the plan either looks
like bAc or cAb. Reflecting in a mirror if necessary, we can assume it is bAc. b
must be flanked by a lady, and the only possibility is C, while c must be seated
next to B, giving CbAcB. Finally a takes the remaining seat: aCbAcB. That’s fine
at dinner, but at breakfast it has every husband sitting opposite his wife,
which is not permitted.
So there’s no answer? Not at all. Billy O’Nair is far too resourceful, and
also too hungry. Nobody said that there had to be just these three couples at
the table. So Millie (D) and Billy (d) joined them, bringing their
place-settings and chairs with them. There was plenty of room because, as I
said, it was an enormous table. Then they arranged themselves in the order
AcDaBdCb, say. Note that with four couples, gentlemen sit opposite gentlemen and
ladies opposite ladies anyway. Deb ended up with a table all to herself, which
suited everybody fine. -
The key is to work backwards. Coachload 10 took half of what there was,
plus half a swimsuit, to leave one. That is, the final one plus half a swimsuit
amounted to half of the total before coach 10 arrived. So the total must have
been three. By the same argument coachload 9 started with 7, coachload 8 with
15, coachload 7 with 31. You can go on like this for all ten coachloads, or you
can observe that the numbers are one less than a power of two. Either way, when
the first coachload of tourists arrived there were 2047 swimsuits in the shop.
So they had sold 2046. -
Six changes of direction suffice. The figure shows the opening
moves—the remainder are similar. The red train is split into three
sections: the engine and 7 carriages, 8 carriages, and 1 carriage. The blue
train remains complete throughout.
First the blue train pushes 9 (the block of 8, plus the 1) carriages of the
red one on to the eastern line, leaving the engine and 7 carriages on the south
loop. It drops the end carriage, and pulls the other 8 onto the south loop while
pushing the red engine and its 7 carriages on to the western line. Then it
returns via the north loop, picks up the lone red carriage, and pushes the
remaining 8 red carriages round the south loop to join the rest, dropping off
the lone carriage on the loop as it passes.
Finally the blue train returns once more via the north loop, reverses on to
the south loop to push the last red carriage on to the western line to join the
rest of the red train, and proceeds back across the loop to the eastern line.
Both trains are now free to proceed—and so is the O’Nair car.
