Some Interesting Mathematical Puzzles.........

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 to puzzle : Folded sheet of paper

Draw a straight line between the two corners used to make the fold, and consider similar triangles.

------------------------------------------------------------------------------------------------------------

 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.)
------------------------------------------------------------------------------------------------------------
 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 to puzzle 5: Confused bank teller

Let x be the number of dollars in the check, and y be the number of cents.  Using the fact that one dollar is equal to 100 cents, write a Diophantine equation connecting x and y.

-----------------------------------------------------------------------------------------------------------

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 to puzzle 6: Ant on a box

The shortest path is a straight line over the top of the box.