How would you divide 9 pitas equally amongst 10 people?
Let kids come up with ideas
Here's an idea: Initially, split 5 pitas into halves and give each one a half. Now divide the rest into thirds, and give each one a third. Finally divide the remaining 2/3 into 1/15ths and give each one a piece. So 9/10=1/2+1/3+1/15
Use this method to divide 7 pitas amongst 13 people
Let kids give two numbers, and lets try with that
Question: Is it always possible to divide in this manner. Specifically, can you always divide n pitas amongst m people, so that the numerator of each term is 1?
For m/n, what is the largest first fraction? 1/ROUNDUP(n/m)
Now if you subtract the two, what are you left with?
Progress through these steps. What do you notice about the numerator progressively?
The numerator is strictly decreasing. What must happen eventually? It should go to 1
Now can we formalize a proof
What method can we use (Younger kids may not remember mathematical induction)
Base case: m=1, then m/n is already in that format
Induction Hypothesis: Assume that this can be done for all m<=k
Induction Step: Prove that this can be done for m=k+1
There is a non-uniform rope which burns completely in one hour if lighted at one end. If lighted at both ends, it would burn in half hour. Of course, you can also burn the rope in 3 places (2 ends and one in between somewhere) to burn it faster.
How do you measure 45 minutes
Answer: Burn a rope at both ends, and another rope at one end. In 30 minutes, the first rope would burn out - light the other end of the second rope at that time, and it will burn out in another 45 minutes
How do you measure 40 minutes
Answer: Burn a rope at both ends to measure 30 minutes. Now light the second rope at both ends and two place in middle, thereby creating three ropes simultaneously burning at both ends. Whenever a subsegment burns out, light another existing subsegment in the middle - so you always have three ropes burning at both ends. This will create 10 minute measure. Note that this is equivalent to Egyptian fraction 2/3=1/2+1/6
Can you devise an alternate solution to 45 minutes using this logic. Note that 3/4=1/2+1/4
Can you think about a way to measure any unit? Note that denominators were even in both the examples above. How would you deal with odd denominators?
Answer: You can treat odd denominators as sum of two even denominators - 1/3=1/6+1/6
Can you think about a solution for measuring 50 minutes?
Answer: Measure 40 minutes like before, and another 10 like the second rope in that solution
Homework Problem:
(MartinColl Page 203) You know how to place 8 queens on a chessboard so that no two attack each other. Can you place 10 queens so that each queen attacks exactly one other queen?