Monday, May 24, 2010 (Victoria Weekend)

The first two problems are motivated from "Our days are numbered" by Brown.

1. Here is a simple game about mind reading:  Bob tells Alice to guess two numbers between 0 and 9.  Bob asks Alice to perform the following four steps (a) multiply the first number by 5 (b)  add 9 to the product (c) double the sum (d) lastly add the second number to the result. Then he asks Alice to tell him the result - and then he subtracts 18 from her answer, and now he knows both the numbers which Alice guessed! Why it works? (For example if Alice guessed 3 and 5, then the steps results in (a) 3x5=15 (b) 15+9=24 (c) 2x24=48 (d) 48+5=53. Bob subtracts 18 from 53 and it equals 53-18= 35 - the two digits which
2. Show that the remainder, when a number is divided by 9, is the same as the sum of its digits (if the sum exceeds 9, then keep taking the sum of the digits, till it is below 9). For example, we know that 345/9 results in a remainder of  3. The sum of the digits of 345= 3+4=5=12, which in turn is 1+2= 3, and it is the same as the remainder.
3. Continuing on the theme of the angle bisector from the last week, how will you draw two lines which are perpendicular to each other (only compass and ruler).
4. Give a line L and point p (p is not on L), how will you find the closest point to p on L geometric construction..
5. Given a triangle ABC, how will you draw an incircle of ABC. An incircle is the largest circle completely contained in the triangle. By the way the center of this circle is the meeting point of the angle bisectors.
6. What about circumcircle of ABC - that is the smallest circle that completely contains ABC. Its center is on the perpendicular bisectors of the  sides of triangle.