Notation:   a^b means a raised to the power b.      E.g.  a^2 is a squared.

 1. THE BARN AND THE CHICKEN COOP     (posted 1991)

      One rainy evening, a farmer asked his son "Do you know how far it is
 between our barn and our chicken coop?  I want to run a two-by-four from one to
 the other and I need to know exactly how long to cut it." The son replied, "No,
 but I noticed once that when your 24 foot ladder was leaning against the barn
 with its foot against the coop, and your 16 foot ladder was leaning against the
 coop with its foot against the barn, the two ladders crossed at a point exactly
 as high as your head.  We should be able to figure it out from that." The
 farmer was 6 feet tall.  How far apart were the barn and the chicken coop?

2.  TRIANGLE AREA FROM ALTITUDES    (Posted Sept 1994)

     If the three altitudes of a triangle are a, b, and c, what is its area A?

 3.  BASEBALL COACH    (posted 1991)

     A baseball coach has to select a player to play shortstop, and seeks to
find which of his N players makes the most consistently quick and accurate
throws.  To do this, he has his first two players, A and B, stand on bases 90
feet apart and throw the ball back and forth as many times as they can in two
minutes.  Dividing the two minutes by the number of throws he computes an
throwing time S1, which is the sum of the time per throw for players A & B:

                     TA + TB = S1

He does the same for players B&C, C&D, D&E, ... M&N, and N&A:

                     TB + TC = S2 : :
                     TM + TN = SM TN + TA = SN.

He solves these equations to get the individual throwing times TA, TB,  etc.
What is TA for an even number of players?.  For an odd number?

6.  AVERAGE MILEAGE:  (posted 1991) The EPA mandates that the cars sold by each
manufacturer must average at least 28 mpg.  The Dreamboat Auto Co. produces
luxury cars which get 10 mpg.  To satisfy the EPA, they propose to raise the
price of their cars from $50,000 to $60,000, and give a Hando, which gets 50
mpg, to each purchaser.  Now each buyer buys two cars, which get an average of
(50+10)/2 = 30 mpg.  The EPA disagrees.  Why?

7. SAME TEMP AT ANTIPODES  (posted 1993) Prove that if the temperature on the
surface of a sphere is continuous, then there exists at least one pair of
opposite points for which the temperature is the same.

8. BUG HEADS NORTHEAST:  (Posted 1993) Assume that the earth is a perfect sphere
40000 km in circumference.  A small bug flies from the intersection of the prime
meridian and the equator, heading always 45ø East of north.  When he is one mm
from the north pole, what is is longitude to the nearest milli-degree?

10.  SNAIL ON A RUBBER BAND.  (Posted March 1994) The left end of a rubber band
one kilometer long is fastened at position x = 0.  At x=0 and time t=0, a snail
on the rubber band starts crawling steadily to the right at 1 cm/sec.  At t=1
second, the snail has crawled to x = 1 cm, and the rubber band is suddenly
stretched to 2 km by moving its unfastened right end, carrying the snail with it
(to 2 cm from his starting point.)  At t=2 seconds, the snail is at 3 cm, and
the rubber band is suddenly stretched again to a length of 3 km, carrying the
snail to x = 4.5 cm.  The process of crawling and stretching continues:  The
snail crawls steadily at 1 cm per second, but at the each second the rubber
band's length is suddenly increased 1 km.  Does the snail ever reach the right
end of the rubber band?  If so, when and where?   --From a Martin Gardner column

12.  2-FOOT HIGH CORED APPLE:  (Posted Feb 94)  A hole is drilled into a perfect
sphere, creating as "cylindrical tunnel" which runs from pole to pole.  [That
is, the axis of the hole coincides with a diameter of the sphere.]  After the
hole is drilled, the remaining pat of the sphere is precisely 2 feet tall.
[That is, its height, measured parallel to the drilling axis, is 2 feet.]  What
is the volume of this remaining portion?  (Courtesy Scott Stevens)

13.  COINCIDENT BIRTHDAYS:  (Posted 1994)  Addressing N strangers, you say "I
bet that at least two of you have a birthday on the same day of the year."  How
large must N be for this to be a favorable bet?

16.  JOSEPHUS' SURVIVAL TRICK: (posted 1993)
Here's a cute puzzle submitted by Deana Hoisington.

Here is the warm up to understand what is going on...
12 jurors are in a room.  A man with 11 bullets breaks in and starts
shooting people.  His shooting pattern is as follows:
Shoots a person, skips a person, shoots a person, skips a person...

i.e. for 12 jurors this is the order he will shoot them in:
    (you may want to follow on a piece of paper.)

1 is killed, 2 is skipped, 3 k, 4 s, 5 k, 6 s, 7 k, 8 s, 9 k, 10 s, 11 k, 12 s
2 k, 4 s, 6 k, 8 s, 10 k, 12 s, 4 k, 8 s, 12k.

Juror 8 is left to live.  Now for the general question...

Given N jurors in a room and a killer with N-1 bullets, which juror do you
want to be?  N can be a number from 2 to infinity.

Note added by JWR:  I think this puzzle is of historical interest.  Josephus
used the answer to survive a mass suicide of rebelling Jews besieged by a
Roman army.

17.  OH WHERE, OH WHERE HAS MY LITTLE DOG GONE:  (Posted 1993)
    A student, a teacher, and a dog all start walking at the same time from the
    same point, in the same direction.  The student walks straight forward at
    5 mph, the teacher walks straight forward at 3 mph, and the dog runs back
    and forth between them at 10 mph, taking no time to turn around.  After
    one hour, where is the dog, and what direction is he facing?

18. MAGNA CARTA: (posted 1994)
    "Bring meat" they cried, "bring wine, bring mead!"
    "The Charter's signed and all's agreed."
    King John with all his barons drinks
    Each once with each his goblet clinks,
    A bell, each goblet peals a chime
    To herald freedom in their time.
    That's what they say.  'Twas long ago
    But here there's something we should know.
    Those clinks all totalled nine-nought-three.
    How many goblets there d'you see?

19.  FIND THE COUNTERFEIT COIN  (Posted 1994)

This is the most perplexing tale,
Of how to use a double-pan scale.

Twelve coins are the same to normal sight
But one may be false, either heavy or light.

And though it may sound fairly tough
To find the whole truth, three weighs are enough.

22. ALPHABETIC PRODUCT:    (Posted May 94)
    Here's a quickie, but read carefully:  Find all the values of x which
    satisfy condition (1) and equation (2):

    (1)  Equation (2) contains a product of 26 factors, one term corresponding
    to each letter of the alphabet.  x is an unknown real number, and  each of
    the other letters stands for some other real number.  No two of the 26
    letters stand for the same number.

    (2)    (x-a)(x-b)(x-c)...(x-z) = 0

25.  IN THE HOBBY SHOP   (Posted August 94)

    "I was in Sam's store today,", said Jim. "He told me that our kids had been
in earlier and bought a model airplane kit for $13.50."

    "Well!"  said Edna, "Where did they ever get the money?"

    Jim laughed.  "They got it from their piggy banks.  Sam was still laughing
about it.  James had coins all of one denomination, and Peter had all coins of
another.  Sam said that each child arranged his coins in rows of piles--as many
rows of piles as there were piles to a row, and as many piles to a row as there
were coins to a pile."

  What were the denominations, and how many coins of each were there?

32.  (Posted fall 1994) CACHING ACROSS THE DESERT:  A desert traveller is
marooned at an oasis 50 miles from the nearest town.  At the oasis he finds a
spring and a large number of empty gallon jugs.  He can carry two jugs, and can
hike for 10 miles on a gallon of water, or 5 miles on a half gallon.  He can
walk at 5 miles per hour.  How many hours of hiking does it take him to cross
the 50 miles to the town?

34.  A NOT-SO-WISE WISEMAN: (Posted 1996) In his book 1,2,3...Infinity, George
Gamov tells this story of the origin of the game Chess.  Supposedly an advisor
to a middle eastern king invented Chess, and presented it to the king.  The
king, delighted with it, offered the inventor any reward he wished.  This offer
worried the king, but when the inventor named his reward, the king was relieved
over how little was asked.  But when the time came to pay the reward, the king
had the inventor beheaded.  Here is what the inventer requested:

A chess board has 64 squares.  Place one grain of wheat on the first square,
then double this amount on the second square, then double it again on the third
square, and continue with the doubling until all 64 squares have wheat on them.
This wheat is the chosen reward.

A grain of wheat weighs roughly four milligrams, and the PRESENT annual world
wheat production is 500 million tons.  (1 ton = a million grams).  How long
would the world's farmers have to labor to satisfy the inventor's request?

36.  (posted fall 94) A QUICKY FROM THE OLD FARMER'S ALMANAC:
Bignose Bill escaped from the Medicine Bow county jail and headed down the road
towards Cheyenne.  One hour later, the sheriff's deputy discovered the escape and
pursued on horseback.  Forty minutes later the sheriff followed in his police
cruiser.  The deputy travelled 6 mph faster than Bignose, and the sheriff
travelled 44 mph faster than the deputy.  They caught Bignose at the same time.
How far had Bignose travelled?

38.  (Posted 12/94)  I'm writing the puzzle on Dec 26, 1994.  This being the
Christmas season, in the song "The Twelve Days of Christmas", how many gifts did
my true give to me?  That's too easy.  Suppose rather than 12 days, the gift
giving went on for 12 years.  With three leap days, that's 4383 days.  THEN, how
many gifts would be given?  (This can be solved in about 15 minutes with a ten-
digit pocket calculator.)

41. [posted feb 1995]  A pilot on a training flight, takes off from an airbase
in his nation's capital.  He flies exactly 200 miles north, then 200 miles east,
then 200 miles south, and then 191.8 miles west back to his starting point.
Assume the earth is a sphere 24,900 miles in circumference.  What was the
LONGITUDE of his starting point?  (You'll need an atlas or a globe to solve this
one.)

44.  A river flows with velocity v.  A moterboat travels at a speed u relative
to the water, and passes a floating log.  It then travels upstream for 1 hour,
turns around and travels downstream, passing the log 1 km downstream from the
point of the first encounter.   How fast is the river flowing?  (From Ray
Serway's Physics book)   [posted March 1995]

45.  QUICKY but TRICKY:  (posted Fall 95)  Suppose there is a long, thin
election district, essentially strung out along a nearly straight highway (like
some of the gerrymanders in the southeastern states.)  The congresswoman of such
a distrIct wishes to site her office in the district in such a place that if
everyone in the district had to travel from their homes to her office, the total
distance travelled would be minimized.  In terms of the distribution of people
along the highway, describe concisely where she should site her office.

47.  Posted June 1995       THE MORE THINGS CHANGE, THE MORE THEY STAY THE SAME.

     Here's a simple game you can play with a calculator.  Sometimes when you
repeat the same operations over and over, the result becomes a constant.
For example, hit 0, and then repeatively hit the sequence (+ 3 = ln).
Eventually the result will settle on 1.505241496

What you have just shown can be written
ln(3 + ln(3 + ln(3 + ln(3 + ...)))) = 1.505241496,
where " ... " means repeat this sequence forever.

It's tougher to solve the problem in reverse.

Find x where    ln(x + ln(x +ln(x + ln(x + ... )))) = 1.1.
(Give the answer to 10 digit accuracy.)

48.  LEAP YEARS  (Posted July 1995)

    Is the year 2000 a leap year?
    The rule is that years divisible by 100 are not leap years, unless they are
divisible by 400, so the year 2000 is a normal leap year.

Question #1:  How many days are there in the average calendar year?

Question #2:  The number of days in the solar year (equinox to equinox) is
365.242190.  Given this and the answer to Question #1, how long will it be until
the calendar year and the solar year are shifted 180 days relative to each
other?

49.  BEGIN THE BEGAT:  (A computer problem) If there were abundant food, and no
disease or war, how fast could the human population increase?  Assume that each
woman could have a baby every year from ages 14 through 50, and that people live
to age 90.  Assume births are equally divided between sexes.  If you started
with a population of two 14-year olds, how long would it take to reach the
earth's present population of 5 billion people?  (Posted January 96)

50.  THE 1995 FACTOR: (Posted summer 1995) N is a number ending in the digits
1995.  If you discard these four digits, you are left with a number L such that
N is an integer multiple of L.   How many possible values of N satisfy this
condition?  (From Quantum).

51.  IMAGINE THIS:  (Posted Sept 95) A,B, and C are the corners of a triangle.
The line AD bisects angle A, and point D is on the line BC.  The line BE bisects
line AD and point E is on line AC.  What is the ratio |EC/AE|?  (From Quantum.
This one is tough.)

54. COUNT THE SQUARES:  (Posted fall 95)  How many squares are there on a
standard 8x8 checkerboard?  That's too easy.  How many are there on an NxN
checkerboard?

55.  THE VALLEY CAMPAIGN:  (Posted Nov 1995)
   (A calculus problem)  The year is 1861.  Stonewall Jackson is retreating
   down the Valley Pike, north of Harrisonburg, before Fremont's advancing
   forces in the Valley Campaign.  He desperately needs Turner Ashby's Cavalry,
   and sends a courier to bring them up as quickly as possible.  Ashby is
   camped 8 miles west of the Valley Pike near Mt. Solon, where the Valley Pike
   runs in a straight line south.  The courier can average 15 mph on the Valley
   Pike, and 9 mph overland.  How far north of Ashby's camp should he cut away
   from the Pike and head southwest overland, in order to reach Ashby in the
   shortest time?

  56.  REDUCING THE DEFICIT  posted Sept 95
The US national debt right now is five trillion dollars, and the annual deficit
is 320 billion (US Treasury figures).  The treasury is auctioning off five-year
T-bonds at 5.8%.  There are a hundred million U.S. households.  If, as the
Republican Congress is attempting to do, the deficit is linearly reduced to zero
over a seven-year period, in seven years, what will be the national debt, the
debt per household, and the annual interest payment per household, if the
interest rates hold steady?

57.  OVERLAPPING CIRCLES:  (Posted March 96)
Two co-planar circles of radii a and b, with a>b, partially overlap.  Their
centers are separated by a distance x.  Give an expression for the area of
overlap in terms of a, b, and x.

59.  TOO FEW ROOMS:  (posted April 95)
  One wet and rainy April First, to a rustic small hotel
  Came ten foot-weary travellers, that rainy night to dwell.
  The Inn had but nine rooms to spare, each had a single bed.
  "Too many reservations!" the red-faced owner said.
  "Don't worry", said his clever wife, "This problem I can fix."
  "We'll keep them warm and happy with some April Fool's tricks."
  Two guests she entered in room A, and said "I'll be right back."
  The third she put in B, and he jumped right in to the sack.
  The fourth and fifth in C and D, the sixth she took to E.
  The seventh and the eighth she quickly lodged in F and G.
  The ninth she led to roomlet H, then back to A did fly,
  And there she got the tenth and last and led him straight to I.
  Ten travellers slept that April first in the small nine-room hotel.
  And now the joke's on you.  How did she do this trick, pray tell?

61.(Posted May 96)  TRIANGLE TOUGHIE:  This is a challenge for faculty as well
as students.  It can be done by solving a cubic equation, (dropping extraneous
roots), but there is a simpler more straight-forward way as well.  (Hint:  Use
your imagination.)

  Inside an equilateral triangle of side x is an arbitrary point which is at
distances A, B, and C from the corners of the triangle.  Find x in terms of A,
B, and C.

62.  Monkey Problem,  11/21/02 (courtesy Steve Whisnant)

  A monkey and the monkey's uncle are suspended at equal distances from the
floor at opposite ends of a rope which passes through a pulley.  (Assume no
friction and equilibrium).  The rope weighs four ounces per foot.  The weight of
the monkey in pounds equals the age of the monkey's uncle in years.  The age of
the monkey's uncle plus the age of the monkey is four years.  The monkey's uncle
is twice as old as the monkey was, when the monkey's uncle was half as old as
the monkey will be, when the monkey is three times as old as the monkey's uncle
was, when the monkey's uncle was three times as old as the monkey was.  The
weight of the rope plus the weight of the monkey's uncle is 1.5 times the
difference between the weight of the monkey's uncle and the weight of the monkey
plus the weight of the monkey's uncle.

  How long is the rope, and how old is the monkey?

 4.  QUADRILATERAL IN A PARALLELOGRAM:  Prove that if a quadrilateral of area A
is inscribed in a parallelogram of area 2A, then at least one diagonal of the
quadrilateral is parallel to a side of the parallelogram.  (From Quantum)

5.  Show that if x and y are non-negative integers, then any non-negative
integer n may be expressed as ((x+y)^2 + 3x -y)/2, with unique values of x and
y.  (From Quantum)

9.  INTERSECTING CIRCLES.  Three circles of radius 1 cm on a plane sheet of
paper intersect at a common point.  Prove that the other three points of
intersection also lie on a circle of radius 1 cm.

11.  TANGENTS INTERSECT ON A LINE.  On a plane piece of paper, draw three
non-intersecting circles of different sizes.  For each pair of circles, draw the two
straight lines which are tangent to the pair of circles and which do not pass
between the centers of the pair of circles.  The two tangents of each pair of
circles will intersect at some point.  Prove that the three intersection points
lie on a straight line.

20.  HATS AND CHECKERS

    The help wanted ad said:  "Talk show host wanted.  Must have a quick mind."
Three people applied, passed preliminary interviews, and were told to come for a
simultaneous competitive interview.  They sat togther in a waiting room, and
after some discussion, it was apparent to each that all were intelligent.  The
interviewer then entered with three top hats, and placed one on the head of each
applicant.

    She then said, "I will now turn out the lights and place a red or black
checker on each hat.  When the lights come on, if you see a red checker, raise
your hand.  The first of you to tell me the color of the checker you're wearing
gets the job."

    When the lights came on, all three raised their hands.  After five seconds,
One of the three applicants said "Red."

     How did the applicant know?  How many checkers were red?

21.  Matched Cards

    Shuffle each of decks of 52 cards.  Turn up a pair of cards, one from each
suit, and see if they match (have the same number of spots). Repeat for all 52
pairs.  What is the probability that at least one pair will match?

23.  Urijah and the Silver Staters:

   Urijah the Hittite came home from the market, whistling and smiling, and
kissed his wife Hannanna.  "We had a good day."  he said.  Serve wine with the
dinner tonight.  I sold all thirty-eight of our calves for forty silver staters."

     At dinner that night with his wife and son, he recounted his success.  His
son Dumuzi was his pride and his hope.  Dumuzi studied each day in the temple,
and was learning to read and write the tablets.  Dumuzi could count and cipher,
and perhaps someday would be a scribe, priest, merchant, or vizier.  But today
was Urijah's day.  "I myself bargained with the Grand Vizier of the Great King,
may he live forever. He wanted to pay 35 staters for 38 calves, and I demanded
50.  Finally we agreed on 39 staters for 37 calves."

    "Who bought the other calf?"   asked Hinnana.

    "I sold it to Hanno the Phoenician, for one stater."

    "Where is the coin Hanno paid?  asked Dumuzi.

    "In the chest, with the other thirty-nine" replied Urijah.  Why do you ask?

    "Because after you left the market, the Grand Vizier arrested Hanno for
paying his tax with a counterfeit stater.  He was tried on the spot, found
guilty, and the soldiers of the Vizier cut off his right hand."

    "Oh!"  shrieked Hannanna.

    "It is just."  said Urijah.  "To pay with a bad coin is robbery.  And Hanno
was a fool.  Everyone knows the Grand Vizier weighs every gold and silver coin
which crosses his counter.  He could get away with that in Ugarit, but not in
our city."

    "But what of your coins, father?  Hanno may have given you a bad coin.  You
must find out if one is bad and destroy it, lest you lose your hand too."

    The next day, Urijah and Dumuzi went to the palace with their forty
staters, and explained the situation to the Vizier.  The Vizier told them that
the counterfeit coin would be either too heavy or too light.  They would be
allowed to use the royal scales to check the coins, but for each weighing the
charge was one dinarius.  The Vizier would allow them to use one of his own
staters as a known weight against which to measure the others.  For this there
would be no charge.

    Urijah's face fell.  "Perhaps the first coin I weigh will be bad, so I will
lose one dinarius to find it.  But if the last coin I weigh is bad, or if they
are all good, then it will cost me forty dinari!  Forty dinari, just to use the
scales to find one bad coin?  And perhaps none is bad at all!  This is
insufferable.  It is like losing my whole fig harvest, or my whole flock of
doves.  But I have no choice."

    "Wait!" said Dumuzi.  "Let us go home today.  I think if we are clever we
will not need to use the royal scales forty times.  If I can save you some
dinari, will you give me half the savings?"

    "Of course, my son." said Urijah.

     The next day Urijah and Dumuzi returned to the palace with the forty
silver staters, forty dinari, and four bowls.  One bowel was marked "Heavy",
one "Light", one "Unknown", and one "Good".  "Now oh, Grand Vizier of the Great
King, may he live forever, I pray you to loan me your stater, and I will find
if Hanno has robbed me.  Dumuzi has a way to find if there is a bad coin, and
which one it is, in just four weighings." said Urijah.  Dumuzi placed several
staters on the scales at each weighing.  After the first weighing, they
returned the Grand Vizier's coin to him.  Sometimes the left pan fell,
sometimes the right, and sometimes the scales balanced.  There was much moving
of coins between the bowls.  The Grand Vizier watched in fascination.  After
four weighings, they had the answer.

     Urijah paid the Grand Vizier four dinari for using the scales four times
and respectfully thanked him.  He gave Dumuzi eighteen dinari for his
cleverness, and put the other eighteen back in his purse.  Then he went off to
the silver-smith to spend the eighteen dinari he had saved on a silver brooch
for Hannanna.

     And the very next day, Dumuzi was appointed Deputy Grand Vizier of the
Great King, may he live forever.

    How did Dumuzi find the bad coin?  Show how to check forty coins using a
double-pan balance and one known good coin in just four weighings.  Find out if
any coin is bad, and if so, which one it is and whether it is heavy or light.

26.  MONKEYS AND TYPEWRITERS

    "Given enough monkeys with typewriters, they'll eventually write all of
Shakespeare's plays."

     Assume a particular play is 50,000 characters long, and 32 characters are
needed to write it (the alphabet plus 6 punctuation marks).  If you let N
monkeys hit 50,000 typewriter strokes each, how many monkeys are needed for
them to probably write the play?

29.  A BIG Number.

     In an ideal gas, the gas atoms never collide, being points, so
their position in a container is random.  Suppose a container holding 10^24  gas
atoms is divided mentally into a left and right half.  What is the probability
P that exactly 59% of the atoms are in the left half, and 49% are in the right
half?

    This number is a very small number, so its inverse, Q = 1/P is very large.
Q is so large that if you wrote it out as a decimal, with a spacing of one digit
per millimeter, the decimal point would be a very far from the most significant
digit.  How far?

31.  THE LONG VIEW:  The Sears Tower in Chicago is 0.295 miles high.  The earth
is a sphere whose radius is 3957 miles.  If light travels in a straight line
from the beacon on top of the tower to a boater on Lake Michigan, how far away
can the boater be and still see the beacon? (Neglect the height of the boat and
the waves.)

33. FOUR BUGS, two male and two female, are on the corners         M   10"   F
of a 10" square, as shown at the right.  At the same time
each starts crawling at 0.5" per second towards the next bug               10"       10"
clockwise around the square.  Each bug remains headed towards
the next, so their paths curve toward the center, with all                       F   10"   M
four paths meeting at the center of the square.  When the
bugs meet, how far has each travelled?

35.  A hollow tetrahedron is constructed from four equilateral triangles.  The
material is stiff (like glass or masonite), and its thickness is negligible.
A standard marble is 5/8" in diameter.   If the triangles measure 10.2" on each
edge, how many marbles will the tetrahedron hold?

37:  Given a sheet of aluminum foil 10" on a side:  Fold it into a tray with
a square base and sloping sides, with maximum volume.  Find this volume to 6-
digit accuracy.

39.  LET'S GET TOGETHER:   A Virginia Tech graduate designs and builds a
conventional clock, with an hour hand, a minute hand, and a sweep second hand.
He aligns all three hands pointing straight up, and starts the clock, at noon.
The hour and minute hands run clockwise and keep perfect time.  At exactly what
times do the hour hand and minute hand coincide?

BONUS QUESTION: The engineer now notices that the second hand runs clockwise
as intended, but much too slow, and at a speed such that it coincides with the
hour and minute hand whenever they coincide with each other.  What is the
fastest speed at which the second hand could be running?

Submitted by a Virginia Tech grad.

46.  THE ART OF PAPER-FOLDING:  A piece of paper is typically 0.1 mm thick.
Fold it once, and you have twice that thickness.  Fold it again, and the packet
is 0.4 mm thick.  After three folds the packet is 8 sheets thick or 0.8 mm.  How
many times must you fold it for the thickness to exceed the height of the Empire
State Building (1260 feet)?

52.  A fox sees a hare a disance L away, and pursues it, running at the same
speed as the hare.  The fox runs straight towards the hare at speed v, and the
hare runs at an angle of 60 degrees to the right of the line connecting the
hare and the fox.  How long does it take the fox to catch the hare, and how far
from the fox's starting position will that occur?  (From Quantum)

53.  A SQUARE DEAL:  Let a,b,c,and d be consecutive corners of a square as you
move counterclockwise around the square, and let p be any point in the square.
Prove that if you draw a perpendicular from pa to b, a perpendicular from pb to
c, a perpendicular from pc to d, a perpendicular from pd to a, then all four
perpendiculars meet at the same point.  (From Quantum).

60. NOTCHED CHECKERBOARD
   You are given an ordinary checkerboard, with one-inch squares, and thirty-two
dominoes, each one inch by two inches.  It is easy to cover all the squares on
the checkerboard with dominoes, so that no dominoes extend beyond the board.
Now take some scissors, and cut out from the checkerboard the two squares which
lie on a pair of diagonally-opposite corners and discard one domino.  Prove that
it is now impossible to exactly cover the board with the remaining dominoes.

61.  RISKY ODDS:  In the game of Risk, combatting armies are attrited according
to the outcome of dice rolls.  The attacker rolls three dice, and the defender
two.  The two highest attacker dice are compared to the two defender dice.  If
the attacker's highest die exceeds the defender's highest, the defender loses a
man.  Otherwise the the attacker loses a man.  The second highest dice are
compared similarly.  On the average, what is the ratio of attacker losses to
defender losses?  I suspect this is best handled as a computer problem.

62.  HEADWINDS:  This puzzle has two parts.

(a)  A pilot is flying between two cities a distance D apart, and has to
conserve fuel.  There is a headwind of speed W, and the fuel consumption rate
is proportional to the square of the air-speed, V.  What airspeed completes the
trip using the least fuel?

(b) This is the same as part (a) but now the headwind is blowing from an angle
A measured from the plane's route.  What heading, H, (measured from the direct
route), and airspeed should the pilot use to complete the trip with minimum
fuel use?

62. BAD ODDS:  This is a moderately hard statistics problem, with a conclusion
which is surprising, or even shocking.  The question has two parts:

Part I:  A gambling casino offers the following game.  The house plays against
six players, numbered one through six.  Each player puts up $1000.  A die is
thrown 48 times.  On each throw, each player bets 1/5 of his cash ($1000 +
winnings - losses) that his number comes up on the die.  If it doesn't he loses
his bet, and his cash is reduced by a factor of 0.8 = (1-1/5).  If he wins the
house pays six times his bet, and his cash is increased by a factor of 2.2 =
(1+6/5).  Note that the house is paying 6 to 1 on a 1 to 5 chance of winning.
The die is good, so that on the average, each number comes up the same number
of times.  On the average, how much does the house win?

Part II.  The house modernizes the game, replacing the die with a ping-pong
ball selector, initially loaded with 48 balls--8 each numbered 1 through 6.
Thus each number comes up the same number of times.  Now how much does the
house win on the average?

PHYSICS PUZZLES:

1.  RANGE OF AN ASTRONAUT

     Show that if a space ship accelerates constantly with an acceleration of

                          a =  9.8 m/s^2

                         x = cosh t' - 1,

where t' is the time measured in years by the spaceship's clock to travel a
distance x as measured in light years by the people back on earth.  How many
astronaut years would it take an astronaut to travel a billion light years,
(1/2 accelerating, 1/2 decelerating), and return?  Repeat for ten billion.

2.  RACING CAR SPEED

     A racing car, the AJC Special, with tires whose coefficient of static
friction is 1.6, races around a flat oval track consisting of two parallel one-
mile long straightaways, with a one-mile diameter semicircle at each end.  What
is the maximum AVERAGE speed which AJC can negotiate the track.  Neglect
aerodynamic effects.

3.  ROBIN HOOD (English units of course)

     Robin Hood, the only yoeman to shoot an arrow an English mile, is in
Sherwood Forest,  practicing splitting arrows at a range of 0.4 mile, shooting
due north.  He is  using 1/4 lb arrows, and his bow delivers a kinetic energy to
them of 810 ft-lbs (3ft draw, 300lbs pull, 90% efficient).  Neglecting air
friction, but including the earth's rotation and gravity, how far above and to
the left of the target must he aim?

4.  WATER CUTOFF

     The Harrisonburg water system is fed by a 24 inch diameter cast iron pipe
12 miles long, with 1/2-inch thick walls.  Assume a tensile strength of 15,000
PSI for the iron.  If the flow rate is 200,000 gallons per hour, what is the
speed of the water in miles per hour?  What is the shortest time in which the
water can be turned off without rupturing the pipe?

5.  FORCE IN A CAPACITOR

   A commercial NEC capacitor weighs 15 grams, and is packaged in a cylindrical
   shape 1 cm thick, and 2.5 cm in diameter.  It has a capacitance of 1.0 F, and
   a maximum voltage of 5.5 V.  When the capacitance is fully charged, what is
   the force on the dielectric plates, and what is the pressure on the
   dielectric?  Assume a dielectric constant of 2, and make other reasonable
   assumptions.
  Give the answers in metric tons and atmospheres.  (1 metric ton = 9800 N)