CHAPTER 5

Question 1

7 LINK CHAIN FOR KISS

A wizard in Numerical Analysis has a gold chain with 7 links. A Lady programmer challenges him to use the chain to buy 7 kisses, each kiss to be paid for, separately, with one chain link. what is the smallest number of cuts he will have to make in the chain? what is his sequence of payments ?

• cut link?
• links for a kiss?
• link for following transaction?

Answer 1

1 cut in the 3rd link:
2 links for a kiss
1 link on the second transaction,

3 link for a kiss
2 links on the third transaction

and so on.

Question 2

FORGETFUL PHYSICIST, WHAT TIME WAS IT?

A forgetful physicist forgot his watch one day and asked an E.E. on the staff what time it was. The E.E. Iooked at his watch and said: “The hour, minute, and sweep second hands are as close to trisecting the face as they ever come. This happens only twice in every 12 hours, but since you probably haven’t forgotten whether you ate lunch, you should be able to calculate the time.” What time was it to the nearest second?

what are the two clock times?

Answer 2

2 hrs 54 min 35 sec / 2.91 hrs
or
9 hrs 5 min 25 sec /
9.19 hrs

Question 3

6 VERTICES, 3 VERTICES, POLYGON

The faces of a solid figure are all triangles. The figure has nine vertices. At each of six of these vertices, four faces meet, and at each of the other three vertices, six faces meet. How many faces does the figure have?

faces?
edges?
vertices?

Answer 3

F = 14
E = 21
V = 9

f+v = e+2

Question 4

ATOM SMASHER, ARC AND TANGENT

A new kind of atom smasher is to be composed of two tangents and a circular arc which is concave towards the point of intersection of the two tangents. Each tangent and the arc of the circle is 1 mile long. What is the radius of the circle?

angle?
formula?
radius?

Answer 4

θ = 74° 46.2’
r = 5280 ft / tanθ
r = 1437.45 ft

Question 5

FLY AND SPIDER

A spider and a fly are located at opposite vertices of a room of dimensions 1, 2 and 3 units. Assuming thet the fly is too terrified to move, find the minimum distance the spider must crawl to reach the fly.

distance, shortest?
distance, others?

Answer 5

√18 (shortest)

other: √20, √26

Question 6

π/10

Show that π/10 is a root of the equation 5x^4 - 10x^2 + 1 = 0

what must be subsituted in x?

Answer 6

x = tan (π/10)

note: substitute to the equation in RAD mode

Question 7

FLY IN A 40X20X20 ROOM

In a room 40 feet long, 20 feet wide, and 20 feet high, a bug sits on an end wall at a point one foot from the floor, midway between the sidewalls. He decides to go on a journey to a point on the other end wall which is one foot from the ceiling midway between the sidewalls. Having no wings, the bug must make this trip by sticking to the surfaces of the room. What is the shortest route that the bug can take?

shortest route?
formula?

Answer 7

58 feet

clue:
c^2 = a^2 + b^2
where:
a = 40
b = 20+20+1+1 = 42

Question 8

INSCRIBED CIRCLE IN A TRIANGLE

A circle of radius 1 inch is inscribed in an equilateral triangle. A smaller circle is inscribed at each vertex, tangent to the circle and two sides of the triangle. The process is continued with progressively smaller circles. What is the sum of the circumference of all circles?

Answer 8

5π

Question 9

A FARMER OWNED A SQUARE FIELD

A farmer owned a square field measuring exactly 2261 yards on each side. 1898 yards from one corner and 1009 yards from an adjacent corner stood a beech tree. A neighbor offered to purchase a triangular portion of the field stipulating that a fence should be erected in a straight line from one side of the field to an adjacent side so that the beech tree was part of the fence. The farmer accepted the offer but made sure that the triangular portion was of minimum area. What was the area of the field the neighbor received, and how long was the fence?

Answer 9

A = 939120 sq. yards
L = 2018 yards

Question 10

FIVE POINTS ON UNIT SQUARE

Given five points in or on a unit square, prove that atleast two points are no farther than √2 / 2 units apart.

Answer 10

√2 / 2
just read proof

Question 11

Given a point P on one side of a general triangle ABC, construct a line through P which will divide the area of the triangle into two equal halves

Answer 11

just read proof

Question 12

STARTING FROM PRIME MERIDIAN

A man leaves from the point where the prime meridian crosses the equator and moves forty-five degrees northeast by geographic compass which always points toward the north geographic pole. He constantly corrects his route. Assuming that he walks with equal facility on land and sea, where does he end up and how far will he have travelled when he gets there?

arrives where?
distance?

Answer 12

Arrives: North Pole
distance: √2 * 10^7 meters

Question 13

TRIANGULAR PLOT 855, 870, AND 975

Near the town of Lunch, Nebraska there is a large triangular plot of land bounded by three straight roads which are 855, 870, and 975 yards long respectively. The owner of the land, a friend of mine, told me that he had decided to sell half the plot to a neighbor, but that the buyer had stipulated that the seller of the land should erect the fence which was to be a straight one. The cost of fences being high, my friend naturally wanted the fence to be as short as possible. What is the minimum length the fence can be?

Answer 13

600 yards

Question 14

THREE HARES RACING

Three hares are standing in a triangular field which is exactly 100 yards on each side. One hare stands at each corner; and simultaneously all three set off running. Each hare runs after the hare in the adjacent corner on his left, thus following a curved course which terminates in the middle of the field, all three hares arriving there together. The hares obviously ran at the same speed, but just how far did they run?

Answer 14

Exactly 100 yards

Question 15

sinA = 5/13, TRIANGLE

A scalene triangle ABC which is not a right triangle has sides which are integers. If sin A = 5/13, find the smallest values for its sides, i.e., those values which make the perimeter a minimurn.

values of 3 sides?

Answer 15

a = 25
b = 16
c = 39

Question 16

2 SHEEP AND 1 GOAT TRIANGLE

A one-acre field in the shape of a right triangle has a post at the midpoint of each side. A sheep is tethered to each of the side posts and a goat to the post on the hypotenuse. The ropes are just long enough to let each animal reach the two adjacent vertices. What is the total area the two sheep have to themselves, i.e, the area the goat cannot reach?

Answer 16

two sheep have exactly 1 acre

Question 17

ELLIPTICAL BRIDGE

A divided highway goes under a number of bridges, the arch over each lane being in the form of a semi-ellipse with the height equal to the width. A truck is 6 ft. wide and 12 ft. high. What is the lowest bridge under which it can pass?

Answer 17

6√5 ft = 13.42 ft

Question 18

COWBOY TO STREAM AND TO CABIN

A cowboy is five miles south ot a stream which flows due east. He is also 8 miles west and 6 miles north of his cabin. He wishes to water his horse at the stream and return home. What is the shortest distance he can travel and accomplish this?

Answer 18

8√5 miles

Question 19

SEEING TWO AND THREE SIDE OF PENTAGON

While still at a sizable distance from the Pentagon building, a man first catches sight of it. Is he more likely to be able to see two sides or three?

Answer 19

each of these probabilities must be 1/2.

note:
chance of seeing two sides must be the same as the chance of seeing three sides and since only two or three sides can be seen at once.

Question 20

PIRATE, TREASURE AND TRIANGLE OF TREES

A pirate buried his treasure on an island, a conspicuous landmark of which were three palm trees, each one 100 feet from the other two. Two of these trees were in a N-S line. The directions for finding the treasure read: “Proceed from southernmost tree 15 feet due north, then 26 feet due west.” Is the treasure buried within the triangle formed by the trees?

Answer 20

treasure will lie outside the triangle they form

Question 21

ORIGAMI

An Origami expert started making a Nani-des-ka by folding the top left corner of a sheet of paper until it touched the right edge and the crease passed through the bottom left corner. He then did the same with the lower right corner, thus making two slanting parallel lines. The paper was 25 inches long and the distance between the parallel lines was exactly 7/40 of the width. How wide was the sheet of paper?

Answer 21

24 in wide

Question 22

AZOULI GOING TO RAILROAD

The Ben Azouli are camped at an oasis 45 miles west of Taqaba. They decide to dynamite the Trans-Hadramaut railroad joining Taqaba to Maqaba, 60 miles north of the oasis. If the Azouli can cover 18 miles a day, how long will it take them to reach the railroad?

Answer 22

2 days trip

Question 23

CIRCUMFERENCE OF FOOTBALL

A cross section through the center of a football is a circle x inches in circumference. The football is x-8 inches long from tip to tip and each seam is an arc of a circle 3x/4 inches in diameter. Find x.

Answer 23

20.69 inches

Question 24

A-B-C TRIANGLE PROOF

Let c be the hypotenuse of a right triangle with legs a and b. Prove that if x > 2, then a^x + b^x < c^x.

Answer 24

proving

Question 25

YANG YING YUNG CIRCLE

A yang, ying, and yung is constructed by dividing a diameter of a circle, AB, into three parts by points C and D, then describing on one side of AB semicircles having AC and AD as diameters and on the other side of AB semicircles having BD and BC as diameters. Which is larger, the central portion or one of the outside pieces?

Answer 25

Same size

A = 1/3 π R^2
r is a radius of a circle

Question 26

TRIANGULAR DIAPER

A diaper is in the shape of a triangle with sides 24, 20 and 20 inches. The long side is wrapped around the baby’s waist and overlapped two inches. The third point is brought up to the center of the overlap and pinned in place. The pin is to go through three thicknesses of material. What is the area in which the pin may be placed?

Answer 26

2.5 sq. in.

Question 27

FRUSTUM POT

A coffee pot with a circular bottom tapers uniformly to a circular top with radius half that of the base. A mark halfway up the side says “2 cups.” Where should the “3 cups” mark go?

Answer 27

The three cups mark is about 2% of the way down from the top of the pot

Question 28

7 POINTS ON THE PENTAGON

How can seven points be placed, no three on the same line, so that every selection of three points constitutes the vertices of an isoceles triangle?

Answer 28

• 5 points at the vertices of a regular pentagon
• 6th point at the center of the pentagon
• 7th point above the center at a distance equal to the radius of the pentagon.

Question 29

ICICLE FORMED AND BIGGER

An icicle forming from a dripping gutter is in the shape of a cone five times as long as it is wide (at the top). A few hours later it has doubled in length and the generating angle has also doubled. How does its present weight compare with its previous weight?

Answer 29

33 times heavier than before

Question 30

CAKE

A hostess plans to serve a square cake with icing on top and sides. Upon determining how many guests want cake, what method should she use to insure that each guest will receive the same amount of cake and icing?

Answer 30

• Lays off equally spaced marks around the perimeter of the cake
• The number of marks being equal to the number of guest
• makes a vertical cut from each mark to the center of the cake.

By elementary geometry
* top areas are equal, insuring equal volumes.
* side areas are equal, insuring equal icing areas.

Question 31

MEDIAN OF A TRIANGLE

Prove that each median of a triangle is shorter than the average of the 2 adjacent sides.

Answer 31

proof only

Question 32

LATTICE POINT

Define every point of the plane with 2 integer coordinates (e.g. [3,0] or l-5,21) as a “lattice point.” Let every pair of lattice points in the plane be connected with a “lattice line.” Prove or disprove: “The Iattice lines cover the plane”

Answer 32

proof
* lattice lines do not cover the plane.

Question 33

SIN (X+Y) = SIN X + SIN Y

A student beginning the study of trigonometry came across an expression of the form sin(X+Y). He evaluated this as sinX + sin Y. Surprisingly he was correct. The values of X and Y differed by 10°; what were these values, assuming that 0° < X < Y < 360°?

Answer 33

one must be 175° and
the other 185°.

Question 34

TREASURE BURIED ON A LAKE

Above is a map of Lake Puevigi. The cross represents a buried treasure cache. Cover the right hand half of the diagram. Now answer: “Is the treasure in the lake or on land?”

Answer 34

• In the lake.

note:
It is simple to verify that a point is inside a closed curve if and only if it requires an odd number of “crossings” to be outside. In this case the number is 3.

Question 35

ISOCLES TRIANGLE 2 EQUAL SIDE AND 3RD SIDE

If the equal sides of an isosceles triangle are given, what length of the third side will provide maximum area? (No calculus, please.)

Answer 35

for maximum area

• 3rd side length = √2 times length
  of one of the equal sides

Question 36

TRIANGLE AND INTERSECTION OF CIRCLES

One side of a triangle is 10 feet longer than another and the angle between them is 60°. Two circles are drawn with these sides as diameters. One of the points of intersection of the two circles is the common vertex. How far from the third side is the other point of intersection.

Answer 36

• At zero distance!
• point of intersection will lie on the third side

Question 37

CUBICAL CONFIGURATION

Here’s a rather unusual optical illusion. How many different confIgurations can you “see”?

Answer 37

at least 3

Question 38

TRIANGLE VERTEX ON CENTER OF SQUARE

The isosceles right triangle shown above has a vertex at the center of the square. What is the area of the common quadrilateral?

Answer 38

12.25

Question 39

FLAG INSIDE THE TRACK

There is one flag at the entrance to a racetrack and another inside the track, half a mile from the first. A jockey notes that no matter where he is on the track, one flag is 3 times as far away as the other. How long is the track?

Answer 39

circumference = 1980π
diameter = 1980
track is circular

Question 40

HUMMING FEEDER BIRDS

Through binoculars a bird watcher observed a hummingbird feeder between one and two o’clock of an afternoon. He timed the visits and saw a ruby-throat take a drink at 1,5,6,8, 15, 16, 19,22, 27, 29, 32, 36, 38, 43, 45, 49, 50, 57, and 58 minutes after the hour of one. The last visit he saw took place at two, at which time he left in perplexity. He knew from experience that a hummer’s “feeding cycle” is remarkably stable and is generally between 5 and 15 minutes long. This one seemed rather erratic, to say the least. Can you advise him on what was going on?

how many humming birds?

Answer 40

3 hummingbirds were sharing the feeding station with cycles of 7, 11 and 13 minutes, respectively, in the order in which he first observed them.

Question 41

SHUFFLE BOARD

What is the longest 6’ wide shuffie board court which will fit in a 20’x 30’ rectangular room

Answer 41

30 7/8 = 26.25