Batch 3 - Class 109 - Mathematical Hockey

• (Contributed by Ishaan) In a two player game, each player picks a prime number of pebbles from a given pile. Assuming both players are super smart, who will win the game starting with 105 pebbles in the pile?
• Answer: First person. Pick 101 leaving the other person with 4 pebbles

• Two teams; each player must take turns - can do random or assigned play as long as everyone gets turn
• 5 minutes per problem. Person to answer first wins if answer is right. Else the other person wins without having to answer
• If no one answers for 5 minutes, then 30 seconds for the next person scheduled on the team to answer
• Draw the hockey rink and goal posts and zones
• Display problems on TV screen

• In a particular month of some year, there are three Mondays which have even dates. On which day of the week does the 15th of that month fall?
• Answer: Sunday
• Solve for AB x 4 = CA
• 23 x 4 = 92
• Solve for ABCD x 4 = DCBA
• 2178 x 4 = 8712
• (2 points) I have three special four-sided dice.

  They have one letter on each side.

  When I roll them together I get three random letters which I try to rearrange into a word. In my eight goes so far I have made the words:

  
  

  
  

  CAT, SON, POD, RIG, PEG, TAP, DIN, APE

  
  

  
  

  What are the letters on each dice?
• Answer: CNPR, AGDS, EIOT
• There are two people. The first one says "Both of us are liars". Is the second person a liar or truthteller?
• Answer: Truthteller
• How many steps are required to break a 8x6 chocolate bar into 1x1 pieces - each step is a complete horizontal or vertical break of one piece
• Answer: 47
• The numbers 1, 2, 3, ..., 999999 are all written in a line. What is the number of 1's?
• Instructor Notes: Start with 1-99, and then 1-999 puzzles 
• Pad the numbers to six digits, so 000001, 000002, 000003, ... 999999 - this has 6.10^6 digits, and each digit appears the same number of times! So 1 must appear 60,000 times
• One can also see that 1-9 has 1 "1". Then 1-99 has 1 "1" for each "ten set", and it has another 10 for 10-19. So 20. Similarly, 1-999 has 20 for each "hundred set" and another 100 for 100-199, so total 300. And so on.
• A man is stranded on an island covered in forest.

  One day, when the wind is blowing from the west, lightning strikes the west end of the island and sets fire to the forest. The fire is very violent, burning everything in its path, and without intervention the fire will burn the whole island, killing the man in the process. T

  here are cliffs around the island, so he cannot jump off. 

  How can the man survive the fire? (There are no buckets or any other means to put out the fire)
• The man carries a stick and burns out the eastern end of island before the fire reaches there. Then he relaxes on the eastern end, where the fire can't reach because there is nothing to burn
• (2 points)

  Find the least positive integer n such that every digit of 15 n is either 0 or 8.
• Answer: 8880. Note that if each digit is 0 or 8, then the number should be divisible by 8. Since 15 does not have common factors with 8, n must be divisible by 8. Lets take m=n/8 - then 15 m should be string of 0's and 1's divisible by 3 and 5. The least such number is 1110, hence 15n = 8880, i.e. n = 592
• Identify these logos - Let kids pull out slips numbered 1-20 and identify the corresponding logo head-to-head

• Answer:

• (Dudeney - 120) Farmer Longmore had a curious aptitude for arithmetic, and was known in his district as the "mathematical farmer." The new vicar was not aware of this fact when, meeting is worth parishioner one day in the lane, he asked him in the course of a short conversation, "Now, how many sheep have you altogether?" He was therefore rather surprised at Longmore's answer, which was as follows: "You can divide my sheep into two different parts, so that the difference between the two numbers is the same as the difference between their squares. Maybe, Mr. Parson, you will like to work out the little sum for yourself." How many sheep did Longmore have?
• The farmer had one sheep only! If he divided this sheep into two parts, making one part 2/3 and the other 1/3, then the difference between these two numbers is the same as difference between their squares - that is, 1/3. Any two fractions will do if the denominator equals sum of the two numerators.
• An alternate answer can be that it is any even number, so when you divide it into two equal parts, the difference between these two numbers is the same as difference between their squares - that is, 0.