Batch 3 - Class 192 - Mathematical Hockey (Repeat of Class 109)

Pre-Class Exercise

Find a better solution to the wire crossing problem above

In city A, divide the 100 wires into 50 pairs, connect 49 of the pairs, leaving one pair disconnected. No need to mark these wires. Travel to city B. Use battery + bulb to identify the connected pairs. Label the two wires of the first pair 1a, 1b. For the second pair 2a, 2b, up through 49a, 49b. Label the disconnected pair 50a, 50b.
Connect 1b to 2a, 2b to 3a, 3b to 4a, ... 49b to 50a, leaving only 50b and 1a disconnected. Travel back to city A and identify which of the two free wires makes no circuit with any of the other pairs. This wire maps to 50b on the other side, and the other free wire maps to 50a. Label them as such.
Now take 50a and find the pair it makes a circuit with. Disconnect the pair and find which wire of the pair it connects with. This wire maps to 49b on the other side and it's pair to 49a. Now find the wire that makes a circuit with 49a. This maps to 48b and it's pair to 48a. Repeat until all wires are identified.

Attendance Kushagra, Mahi, Arjun, Aashvi, Rohan, Smiti, Muskaan, Rehan, Rhea

Class puzzles

Explain the rules of Mathematical Hockey

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?

(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?

There are two people. The first one says "Both of us are liars". Is the second person a liar or 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

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 600,000 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. There 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

Homework

(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.

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