Friday, March 30, 2018

3/27 Olympiad #5

As is our routine, I started by going over the problem of the week:

Find six distinct natural numbers A,B,C,D,E,F such that

A + B + C = D + E + F


Once again, I had a girl write a python program to find the solution. I love all the computational computing that is occurring. I need a better way to harness this energy. This time I had her talk about the structure of the loops she used to search the problem space.  In this case, it was a straightforward brute force attack loop over all 6 variables up to a limit and just check for each permutation if the two conditions were met.

This is relatively slow and it produced lots of duplicate solutions as she pointed out that had to be manually disambiguated. I didn't go into it because not enough of the kids have a programming background but there are a few easy improvements that can be made on this approach:

limit = 100

for total in range (6,limit):
    results = dict()
    solns = 0
    for a in range (1,total - 3):
        for b in range (a+1, total - (3 + a)):
            c = total - (a+b)
            if c < b:
            sum = a*a + b*b + c*c
            if sum in results:
                solns += 1
                print "Found %d %d %d and %s total:%d" %(a, b, c, results[sum], total)

            results[sum] = (a, b, c)

    ratio = float(solns) / total
    print "%d solutions for %d ratio=%f" % (solns, total, ratio)

Looping over the sum and then over the 3 digits in increasing order eliminates duplicates and cuts the number of comparisons down significantly.  Note: the key observation is that you only have 2 degrees of a freedom once you've picked a sum for a + b + c.

After this point, we did the final MOEMS olympiad for the year. I delayed this round to fit better with the other activities I wanted to do.  So technically we only had this week to finish and submit the scores.  Two things stood out at me. There was a fraction question that dovetailed with the 2 weeks we've worked on Farey Sequences and Wilf-Calkin trees. Also yet again there was another combinatorics problem that most kids enumerated over rather than calculating a true combination.  Overall, I think the kids scored the highest average of all the rounds. This afforded me the opportunity to have to cold call a few kids that usually are more reticent to demo on the board.  One highlight for me was  a girl proudly getting all the problems right and telling me that she thought this one was easy.  I know she meant relative to the other ones for her but I still had to make a comment to avoid language like "this is easy."  Nevertheless that was  a huge victory. 

I chose a page from the "This is not a math book" for the early finishers with a fun rectangle dividing project. (I brought my box of crayons in for this.)

This was popular but didn't occupy as much time as I expected and I ended up giving out the P.O.T.W early to some kids as well as breaking out my game of 24 cards.


This one comes from Ed Southall and is a fraction  talk activity:

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