Problem Set 1 (10 minutes) NAME::

1. Peter promised to his mom that he will only read 50 more pages of comics before returning to his math assignment. After reading pages 50, 51, … , 100 Peter started on his math assignment but his Mom told him that he broke his promise. Was she right and why?

Solution: Yes, he read 51 pages.

2. The digits of a 3-digit number were written in a reverse order to form a new number. Could this sum of these two numbers be written with just odd numbers and why?

Solution: Yes, for example 429 + 924=1353

3. In ancient Rome, they use the following system for numbers:

I meant 1

II meant 2

III meant 3

IV meant 4

V meant 5

VI meant 6

VII meant 7

VIII meant 8

IX meant 9

X meant 10

XI meant 11

XII meant 12

XIII meant 13

XIV meant 14

XV meant 15

XVI meant 16

In the following equation XIV-XVI=II move one match (every line is a match) so the equality becomes right.

Solution: Move one match from = and get XIV=XVI-II

Problem Set 2 (15 minutes) NAME::

1. The chest full of gold weighs 32 pounds, and the chest that’s half full weighs 17 pounds. How much does the chest weigh?

Solution: 32-17 = 15 pounds is how much half the chest of gold weighs, hence the full chest full of gold is 30 pounds, hence the chest itself is 32-30=2 pounds.

2. Cut the figure into 4 equal pieces.

Solution, e.g.

1. A grasshopper makes 50 jumps on a straight line from the origin. His first jump is 1 feet, second is 2 feet etc 50-th is 50 feet in one of two possible directions along the line. Prove that the grasshopper could not end up in the origin where he started off. (Hint: consider a coordinate line with all even feet ticks colored white and all odd ticks colored black).

Solution: Note that the color of the ticks where the grasshopper lands changes only when he jumps odd number of feet, which he does 25 times out of 50 jumps, since 25 is odd, he will end up on a differently colored tick from the one marking the origin.

Problem Set 3 (20 minutes) NAME::

1. Knights and liars live on an island: knights always tell the truth, and liars always lie, and there are at least 2 knights and at least 2 liars on an island. One day all of them gathered on a central square and each of them pointed to every other person on the square and said either “You are a knight” or “You are a liar”. The phrase “You are a liar” was said 70 times, how many times was the phrase “You are a knight” said?

Solution: If X is the number of liars, and Y is the number of knights, the number of times “You are a liar” was said is X*Y (for all liars addressing knights) + X*Y (for all knights addressing the liars) = 70. So X*Y = 35, and since X>2 and Y>2, X and Y can only be 7 and 5, we don’t know which is which. Liars addressing liars + knights addressing knights is X*(X-1) + Y*(Y-1) = 7*6 + 5*4 = 62 – note that they are not talking to themselves.