鈥淗OLIDAYS are great,鈥 said Millie O鈥橬air, as she rummaged through the last
two of the six suitcases in search of space to pack her best earrings, 鈥渆xcept
for the packing.鈥 She grimaced, levered one of the cases closed, and attacked
the combination locks on the remaining one.
鈥淟et me help,鈥 said her teenage daughter Deb. 鈥淚 love packing.鈥
鈥淥K, Debs. You go and help Dad pack this lot in the boot.鈥 Billy O鈥橬air had
bought the family car 鈥攁 Redundant Redstart鈥攁t an auction, declaring
it to be 鈥渁 robust machine of total reliability鈥. Millie agreed that this
sentence was entirely accurate鈥 except for the syllables 鈥渞o鈥 and
鈥渞别鈥.
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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鈥檚
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鈥攂ut the sixth
wouldn鈥檛 fit. Then they began shouting at each other again.
鈥淎ll right, you two, what鈥檚 the problem?鈥
鈥淒ad keeps putting them in wrong!鈥
Billy glared at her. 鈥淣o I don鈥檛, Debs. There鈥檚 something funny about the
sizes鈥攊t looks as if they ought to fit easily, but they don鈥檛 seem to. All
of the cases are the same鈥 rectangular blocks two feet square and one foot
迟丑颈肠办.鈥
鈥淎nd the boot?鈥
鈥淲ith the shelf removed there鈥檚 an exact 3-foot cube available.鈥
Millie thought for a moment. 鈥淪o 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.鈥
鈥淩ight. But there isn鈥檛. It鈥檚 not just the size, but the shape that
尘补迟迟别谤蝉.鈥
Millie pursed her lips in contemplation. 鈥淢ust be some way to鈥攁ha!鈥
鈥淎ha what?鈥
鈥淵ou鈥檙e going about this all wrong. The reason you can鈥檛 solve the problem is
that both of you are worrying about packing the cases into the boot.鈥
Billy stared at her. 鈥淏ut what else should we be trying to pack in? There
isn鈥檛 anything else!鈥
Millie gave them a Mona Lisa smile. 鈥淥h, but there is.鈥
鈥
It was nearly midnight when the car pulled into the entrance to the Shades
Hotel. At least, that鈥檚 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.
鈥淧ayment in advance,鈥 she told Billy.
It was too late to argue, they all needed a rest. Deb was fractious and
Millie was exhausted. 鈥淟et鈥檚 see鈥 we鈥檙e booked in for five nights,
at鈥攈ow much per night?鈥
鈥淭he price is a whole number of pounds,鈥 said Mrs Grimm, 鈥渨hich exceeds its
reversal by one-eleventh of the digits鈥 total.鈥 She waved her finger at Deb, who
was about to speak. 鈥淎nd the answer is not zero, Miss Clever-clogs. It鈥檚 one of
the little tests I set my guests to keep them on their toes. You鈥檝e got three
minutes to solve it, otherwise the price goes up to 拢100 a night.鈥
Billy groaned.
鈥
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鈥擬r
and Mrs Abbot, Mr and Mrs Bishop, and Mr and Mrs Cardinal. The O鈥橬airs 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鈥橬airs鈥 order. The waitress was looking harassed, and the toast had long ago
gone limp and cold. Finally Billy could stand it no longer. 鈥淓xcuse me, young
lady, but what is this? The Mad Hatter鈥檚 tea party?鈥
She turned round, embarrassed. 鈥淚鈥檓 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.鈥
鈥淥h, for heaven鈥檚 sake!鈥 exclaimed Billy. 鈥淪urely you鈥檝e got enough gumption
迟辞鈥︹赌
鈥淵es, but there鈥檚 another rule, at breakfast only. A husband and wife may not
sit directly opposite each other across the table.鈥
鈥淲hy not?鈥 said Billy.
鈥淗ave you looked at yourself in the mirror this morning, sir?鈥
鈥淥h. Right. But surely it can鈥檛 be that hard to sort out some arrangement so
that Mrs Grimm is satisfied and we can all get served.鈥
鈥
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.
鈥淪wimsuits?鈥 She shook her head. 鈥淪orry, madam, but we don鈥檛 stock swimsuits.
There鈥檚 not much call for them.鈥
鈥淏ut this is a seaside resort. And you鈥檙e a swimwear shop.鈥
鈥淥h, yeah. Um鈥 well, the weather鈥檚 bad.鈥
鈥淣ot today, it isn鈥檛. Anyway, you鈥檝e got one in the window.鈥
The assistant looked blank for a moment. 鈥淥h. That swimsuit. No, you can鈥檛
have that one.鈥
鈥淲hy ever not?鈥
The assistant burst into tears. 鈥淟ook, I鈥檓 sorry, but I had an awful day
yesterday, I can鈥檛 balance the books, I鈥檝e 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鈥檚 extremely proud of it, and he鈥檇 blow his top if we so
much as touched it鈥攅ven to sell something.鈥
Millie鈥檚 heart went out to the poor woman. 鈥淭hen I鈥檒l buy another one, it
doesn鈥檛 matter.鈥
鈥淪orry, but鈥攚ell, you see, we鈥檝e only got the one.鈥
Millie found this hard to believe.
鈥淲ell, 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鈥斺
鈥淗alf a swimsuit? Were they French tourists?鈥
The assistant looked baffled. 鈥淣o, 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.鈥
鈥淥h dear. It鈥檚 the last day of my holiday, and the first one with any
蝉耻苍蝉丑颈苍别.鈥
The sales assistant came to a decision. 鈥淭ell you what, madam. As I said,
I鈥檝e made a bit of a cock-up in the stocktaking department and I don鈥檛 actually
know how many swimsuits there were to begin with. Mr Savage-Brut鈥檚 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鈥檒l think up some excuse to explain why
it鈥檚 gone.鈥
鈥
鈥淪oon be home now,鈥 said Billy, turning off the M99 on to a tiny country
lane. The Redstart鈥檚 motor coughed, faltered, and then picked up again. Billy
breathed a sigh of relief.
鈥淚s this the right road?鈥 asked Millie.
鈥淭here are road works at the A777,鈥 said Billy. 鈥淭his is a short cut.鈥
鈥淒oesn鈥檛 it cross the railway line?鈥 asked Deb.
鈥淭wice,鈥 said Billy. 鈥淭here鈥檚 a loop. But it only takes a few minutes for a
train to鈥 whoops!鈥
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. 鈥淚 don鈥檛 need to ask what the difficulty is,鈥 he said, 鈥渂ut how did
you get into such a mess in the first place?鈥
鈥淚t鈥檚 not my fault,鈥 said the driver of the blue train. 鈥淭he 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
辫濒肠.鈥
鈥淪o why don鈥檛 you just back off and let me through, eh? I鈥檓 very tired and
I鈥檝e only got another couple of miles till I get home. You鈥檙e obstructing the
谤辞补诲.鈥
The driver of the red train shrugged his shoulders. 鈥淪orry, guv, but Southern
Barriers have standing regulations. Once the barrier goes down, it can鈥檛 go up
again until the train鈥檚 gone past.鈥
鈥淵es, and Northern Crossings have a similar regulation,鈥 said the other
driver.
Millie climbed out of the car and her husband explained the difficulty. 鈥淪o
call in a few more engines,鈥 she said, 鈥渁nd they鈥檒l soon help you sort it
辞耻迟.鈥
The two drivers stared at each other. 鈥淗as EasTransCo got any spare
Locomotive Power Units?鈥 asked one.
鈥淣ot since last April when we sold them to a bunch of Chechen
entrepreneurs鈥 WesTransCo?鈥
鈥淣ope, nary a one. Inefficient to keep unused spares, the CEO said at the
last shareholders鈥 meeting in Barbados.鈥
鈥淚 see,鈥 said Billy. 鈥淭hen we鈥檒l have to manage with what we鈥檝e got.鈥 He
counted the carriages. 鈥淵ou鈥檝e 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?鈥
鈥凌颈驳丑迟.鈥
鈥淔ine. Then you鈥檇 better get on with it.鈥
鈥沦辞谤谤测?鈥
鈥淕et these trains past each other, correctly assembled with the engines in
front. And do it with as few changes in the direction of engines鈥 movement as
possible.鈥 The driver gave him an amused look. 鈥淏ut that鈥檚 RailShunt鈥檚
responsibility, not ours.鈥
Billy took out his diary, and consulted it. 鈥淭oday is the third Tuesday of
the month, which means that under the Level Crossing Users鈥 Charter I am
entitled to compensation of 拢15.75 for every full minute you keep me
waiting beyond half an hour. I imagine it will be docked from your wages. And
we鈥檝e already been stuck here for 20 minutes.鈥
The drivers looked at him, then at each other, and rushed off to their
respective trains.
鈥淚 didn鈥檛 know there was a Level Crossing Users鈥 Charter,鈥 said Millie,
impressed.
鈥淒idn鈥檛 you? Well there is, though what it actually offers is a 10p lollipop
voucher for every week you鈥檙e kept waiting.鈥
鈥
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鈥檚 eye. 鈥淪orry, thought
they were both for me. Here, let me get鈥︹
The doorbell rang.
It was Mrs Bung from next door. 鈥淛ust dropped by to see how you are,鈥 she
said. 鈥淥h, the council left a note to say that they鈥檒l be digging up the road
outside starting at 4 o鈥檆lock tomorrow morning. And then they鈥檒l be cutting the
electric off just before breakfast.鈥
She noticed the horrified expression on Billy鈥檚 face. 鈥淣ever mind, Mr O鈥橬air.
I鈥檓 sure you鈥檒l be lovely and refreshed after that relaxing holiday you鈥檝e had.
And the road works should only last for six weeks, so they say.鈥 She turned
towards the front door. 鈥淣ice to get away from it all, isn鈥檛 it?鈥
* * *
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鈥攆or 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 鈥渁b鈥 (which represents 10a + b) and
the reversal is ba (10b + a). The difference between
them is 9a鈥 9b. The total is 11a + 11b
, and one-eleventh of this is a + b, so 9(a 鈥
b) = a + b. Therefore a + b is a multiple of
9. If a = b = 9, then 9(a 鈥 b) = 0,
which won鈥檛 work. So a + b = 9 and therefore a
鈥 b = 1. It follows thata = 5, b = 4, so the
price is 拢54 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鈥檚 fine
at dinner, but at breakfast it has every husband sitting opposite his wife,
which is not permitted.
So there鈥檚 no answer? Not at all. Billy O鈥橬air 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鈥攖he 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鈥攁nd so is the O鈥橬air car.
