%  Algebra puzzles
%  21-110: Problem Solving in Recreational Mathematics, spring 2010
%  Wednesday, February 3, 2010
%  Brian Kell <bkell@cmu.edu>

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\centerline{\titlefont 21-110: Problem Solving in Recreational Mathematics}
\smallskip
\centerline{\bf Algebra puzzles}
\smallskip
\centerline{Wednesday, February~3, 2010}

\bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent (Problems from {\it The Moscow Puzzles\/} by Boris~A. Kordemsky,
edited by Martin Gardner.)

\medskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problemno37. {\it The Price of a Book.}\enspace
A book costs \$1 plus half its price. How much does it cost?

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problemno61. {\it For the Tsimlyansk Power Installation.}\enspace
A factory making measuring equipment urgently needed by the famous Tsimlyansk
power installation has a brigade of ten excellent workers: the chief (an older,
experienced man) and 9~recent graduates of a manual training school.

Each of the nine young workers produces 15~sets of equipment per day, and their
chief turns out 9~more sets than the average of all ten workers.

How many sets does the brigade produce in a day?

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problemno62. {\it Delivering Grain on Time.}\enspace
A collective farm was due to deliver its quota of grain to the state
authorities. The management of the kolkhoz decided the trucks should arrive in
the city at exactly 11:00~a.m. If the trucks traveled at 30~miles per hour they
would reach the city at ten, an hour early; at 20~miles an hour they would
arrive at noon, an hour late.

How far is the kolkhoz from the city, and how fast should the trucks travel to
arrive at 11:00~a.m.?

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problemno70. {\it What Is It?\/}\enspace
A half is a third of it. What is it?

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problemno73. {\it Would He Have Saved Time?\/}\enspace
Our man Ostap was going home from Kiev. He rode halfway---fifteen times as fast
as he goes on foot. The second half he went by ox team. He can walk twice as
fast as that.

Would he have saved time if he had gone all the way on foot? How much?

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problemno76. {\it A Cake of Soap.}\enspace
If you place 1~cake of soap on a pan of a scale and $3/4$~cake of soap and a
$3/4$-pound weight on the other, the pans balance.

How much does a cake of soap weigh?

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problemno79. {\it Misha's Kittens.}\enspace
Every time young Misha sees a stray kitten he picks up the animal and brings it
home. He is always raising several kittens, but he won't tell you how many
because he is afraid you may laugh at him.

Someone will ask: ``How many kittens do you have now?''

``Not many,'' he answers. ``Three-quarters of their number plus three-quarters
of a kitten.''

His pals think he is joking. But he is really posing a problem---an easy one.

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problemno80. {\it Average Speed.}\enspace
A horse travels half his route, with no load, at 12~miles per hour. The rest of
the way a load slows him to 4~miles per hour.

What is his average speed?

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problemno81. {\it The Sleeping Passenger.}\enspace
A passenger fell asleep on a train halfway to his destination. He slept till he
had half as far to go as he went while he slept. How much of the whole trip was
he sleeping?

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problemno83. {\it A Cyclist.}\enspace
After a cyclist has gone two-thirds of his route, he gets a puncture. Finishing
on foot, he spends twice as long walking as he did riding.

How many times as fast does he ride as walk?

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problemno85. {\it Who Is Right?\/}\enspace
Masha had to find the product of three numbers in order to calculate the volume
of some soil.

She multiplied the first number by the second correctly and was about to
multiply the result by the third number when she noticed that the second number
had been written incorrectly. It was one-third larger than it should be.

To avoid recalculating, Masha decided it would be safe to merely lower the
third number by one-third of itself---particularly since it equaled the second
number.

``But you shouldn't do that,'' a girl friend said to Masha. ``If you do, you will
be wrong by 20~cubic yards.''

``Why?'' said Masha.

Why indeed? And what is the correct soil volume?

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problemno217. {\it Mutual Aid.}\enspace
During the rebuilding after World War~II, we were short of tractors. The
machine and tractor stations would lend each other equipment as needed.

Three machine and tractor stations were neighbors. The first lent the second
and third as many tractors as they each already had. A few months later, the
second lent the first and third as many as they each had. Still later, the
third lent the first and second as many as they each already had. Each station
now had 24~tractors.

How many tractors did each station originally have?

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problemno218. {\it The Idler and the Devil.}\enspace
An idler sighed: ``Everyone says, `We don't need idlers. You are always in the
way. Go to the devil!' But will the devil tell me to get rich?''

No sooner did the idler say this than the devil himself stood in front of him.

``Well,'' said the devil, ``the work I have for you is light, and you will get
rich. Do you see the bridge? Just walk across and I will double the money you
have now. In fact, each time you cross I will double your money.''

``You don't say!''

``But there is one small thing. Since I am so generous you must give me \$24
after each crossing.''

The idler agreed. He crossed the bridge, stopped to count his money~$\dots$ a
miracle! It had doubled.

He threw~\$24 to the devil and crossed again. His money doubled, he paid
another~\$24, crossed a third time. Again his money doubled. But now he had
only~\$24, and he had to give it all to the devil. The devil laughed and
vanished.

The moral: When anyone gives you advice you should think before you act.

How much money did the idler start with?

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problemno222. {\it Vera Types a Manuscript.}\enspace
Mother asked Vera to type a manuscript. Vera decided: ``I will type an average
of 20~pages a day.''

She typed the first half of the manuscript rather lazily, at 10~pages a day.
To make up for it, she typed the second half at 30~pages a day.

``See, I did average 20~pages a day,'' Vera concluded. ``Half of $10+30$
is~20.''

``No, you didn't,'' her mother said.

Who was right?

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problemno228. {\it Young Pioneers.}\enspace
Vitya pledges that his brigade of Young Pioneers will plant half the number of
fruit trees the rest of the Pioneers plant. Kiryusha pledges his brigade, the
largest in the detachment, will plant as many trees as the rest of the Pioneers
(including Vitya's brigade).

Their brigades work the last shift simultaneously. The preceding brigades of
the detachment plant 40~trees. Assuming that both pledges are fulfilled
exactly, how many trees does the whole detachment plant?

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problemno233. {\it Jack London's Journey.}\enspace
Jack London tells how he raced from Skagway in a sled pulled by 5~huskies to
reach the camp where a comrade was dying.

For 24~hours the huskies pulled the sled at full speed. Then 2~dogs ran off
with a pack of wolves. London, left with 3~dogs, was slowed down
proportionally. He reached camp 48~hours later than he had planned. If the
runaway huskies had stayed in harness for 50~more miles, London writes, he
would have been only 24~hours late.

How far is the camp from Skagway?

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\bye
