“Puzzles can lead you into almost every branch of mathematics,”
Date : 11.10.2017
To Learn more : http://www.qbyte.org/puzzles/puzzle02.html
1. Folded sheet of paper 
A rectangular sheet of paper is folded so that two diagonally opposite corners come together. If the crease formed is the same length as the longer side of the sheet, what is the ratio of the longer side of the sheet to the shorter side?
Hint - Answer - Solution
2. Triangular area 
In 
ABC, produce a line from B to AC, meeting at D, and from C to AB, meeting at E. Let BD and CE meet at X.
Let BXE have area a, BXC have area b, and CXD have area c. Find the area of quadrilateral AEXD in terms of a, b, and c.
Hint - Answer - Solution
3. Two logicians 
Two perfect logicians, S and P, are told that integers x and y have been chosen such that 1 < x < y and x+y < 100. S is given the value x+y and P is given the value xy. They then have the following conversation.
P: I cannot determine the two numbers.
S: I knew that.
P: Now I can determine them.
S: So can I.
Given that the above statements are true, what are the two numbers? (Computer assistance allowed.)
Answer - Solution
4. Equatorial belt
A snug-fitting belt is placed around the Earth’s equator. Suppose you added an extra 1 meter of length to the belt, held it at a point, and lifted until all the slack was gone. How high above the Earth’s surface would you then be? That is, find h in the diagram below.
Assume that the Earth is a perfect sphere of radius 6400 km, and that the belt material does not stretch. An approximate solution is acceptable.
Hint - Answer - Solution
5. Confused bank teller
A confused bank teller transposed the dollars and cents when he cashed a check for Ms Smith, giving her dollars instead of cents and cents instead of dollars. After buying a newspaper for 50 cents, Ms Smith noticed that she had left exactly three times as much as the original check. What was the amount of the check? (Note: 1 dollar = 100 cents.)
Hint - Answer - Solution
6. Ant on a box
A 12 by 25 by 36 cm cereal box is lying on the floor on one of its 25 by 36 cm faces. An ant, located at one of the bottom corners of the box, must crawl along the outside of the box to reach the opposite bottom corner. What is the length of the shortest such path?
Note: The ant can walk on any of the five faces of the box, except for the bottom face, which is flush in contact with the floor. It can crawl along any of the edges. It cannot crawl under the box.
Hint - Answer - Solution
7. Five men, a monkey, and some coconuts
Five men crash-land their airplane on a deserted island in the South Pacific. On their first day they gather as many coconuts as they can find into one big pile. They decide that, since it is getting dark, they will wait until the next day to divide the coconuts.
That night each man took a turn watching for rescue searchers while the others slept. The first watcher got bored so he decided to divide the coconuts into five equal piles. When he did this, he found he had one remaining coconut. He gave this coconut to a monkey, took one of the piles, and hid it for himself. Then he jumbled up the four other piles into one big pile again.
To cut a long story short, each of the five men ended up doing exactly the same thing. They each divided the coconuts into five equal piles and had one extra coconut left over, which they gave to the monkey. They each took one of the five piles and hid those coconuts. They each came back and jumbled up the remaining four piles into one big pile.
What is the smallest number of coconuts there could have been in the original pile?
Hint - Answer - Solution
8. 271 
Write 271 as the sum of positive real numbers so as to maximize their product.
Hint - Answer - Solution
9. Reciprocals and cubes
The sum of the reciprocals of two real numbers is −1, and the sum of their cubes is 4. What are the numbers?
Hint - Answer - Solution
10. Farmer’s enclosure
A farmer has four straight pieces of fencing: 1, 2, 3, and 4 yards in length. What is the maximum area he can enclose by connecting the pieces? Assume the land is flat.
Answer - Solution
Date : 03/10/2017
1. Crazy cut
You are to make one cut (or draw one line) – of course it needn’t be straight – that will divide the figure into two identical parts.
2. The coloured socks
Ten red socks and ten blue socks are all mixed up in a dresser drawer. The 20 socks are exactly alike except for their colour. The room is in pitch darkness and you want two matching socks.
What is the smallest number of socks you must take out of the drawer in order to be certain that you have a pair that match?
3. Twiddled bolts
Two identical bolts are placed together so that their helical grooves intermesh as shown below. If you move the bolts around each other as you would twiddle your thumbs, holding each bolt firmly by the head so that it does not rotate and twiddling them in the direction shown, will the heads
(a) move inward,
(b) move outward, or
(c) remain the same distance from each other?
4. The fork in the road
A logician vacationing in the South Seas finds himself on an island inhabited by two proverbial tribes of liars and truth-tellers. Members of one tribe always tell the truth, members of the other always lie. He comes to a fork in a road and has to ask a native bystander which branch he should take to reach a village. He has no way of telling whether the native is a truth-teller or a liar. The logician thinks a moment, then asks one question only. From the reply he knows which road to take. What question does he ask?
5. Three squares
Using only elementary geometry (not even trigonometry), prove that angle C equals the sum of angles A and B.
6. Cutting the pie
With one straight cut you can slice a pie into two pieces. A second cut that crosses the first one will produce four pieces, and a third cut can produce as many as seven pieces. What is the largest number of pieces that you can get with six straight cuts?
7. The mutilated chessboard
The props for this problem are a chessboard and 32 dominoes. Each domino is of such size that it exactly covers two adjacent squares on the board. The 32 dominoes therefore can cover all 64 of the chessboard squares. But now suppose we cut off two squares at diagonally opposite corners of the board and discard one of the dominoes.
Is it possible to place the 31 dominoes on the board so that all the remaining 62 squares are covered? If so, show how it can be done. If not, prove it impossible.
8. The two spirals
One of these spirals is formed with a single piece of rope that has its ends joined. The other spiral is formed with two separate pieces of rope, each with joined ends.
Can you tell which is which by using only your eyes? No fair tracing the lines with a pencil.
Infinite Mathematics 