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PARITY #1: NIM, Sums, and Conjectures

Math Circle 3.17.2016

(March 17, 2016)  Rachel co-led this circle of 8-year-olds, and with her it began.


She threw 4 piles of stones onto the ground and said, “We’ll take turns removing stones from piles until there’s just one left,” she explained.  “I’m playing against the team of all of you.”

“Viking chess!” said L.  “Let’s count the rocks, be tactical,” counseled L to the others.  “Ten.  That was the lowest pile.”

“Ooh, I have a tactic!” said B, drawing the group into a huddle.  “We have to figure out how to do this.”  R removed some stones.  The students counted how many she took and challenged her, “Is that your tactic?”  Play continued.  The students didn’t have a system of consensus or taking turns to manage their turn.  Whoever thought of something just did it.  They were playing a learning game.  Actually, R was playing a teaching game, but the kids didn’t know that.  She removed all but one stone.

“We shouldn’t have done that,” observed S.  She won.”

L:  “How do you win?” This question made it apparent that some had dived right into play without knowing the goal of the game, which in this version was to force your opponent to remove the last stone.  They wanted to play again.  Game 2 began, this time the students against my other helper J.

Right away someone said to J:  “This time you go first.”  They were clearly experimenting with strategies.  Play continued.

“I’m the boss!” declared B.

“I’m the boss,” declared L.  The other students didn’t seem to mind these declarations, and having two declared bosses didn’t much hinder teamwork.  The game ended with J winning but the kids getting excited about possible strategies.



I asked all the students, in turn, to grab a handful of wooden cubes with their eyes closed.  “Is the number of cubes you have even or odd?”  Each had a different strategy – some mental, some physical, and B a combination of both as he kept his eyes closed for the whole activity.  “What does even mean?  How about odd?” I asked.  The consensus seemed to be that you can split even numbers into two groups with nothing left over, and odd numbers have one left over.  Time to play devil’s advocate.

“How many legs does a horse have?  Is that even or odd? A spider?  An octopus?  A starfish?”  I drew a starfish on the board.  “I could take an imaginary knife and split this starfish right down the middle and have the same number of legs on each side.  Two and a half.  The same on both sides with nothing left over.  Therefore, I posit that five is an even number.”  This statement really threw the kids.  Immediately they wanted to accept what I said because I’m bigger and older than they are.  But they had their doubts.  Even M, who was really sure based upon physical and mental math which numbers are odd and which are even, said in a questioning tone, “But five is odd?”  I wrote on the board 2 ½ + 2 ½ = 5.  The kids drew starfish and “sliced” them with their markers and got the same result.  No one was sure.

“This is what I hate about math,” said L.  What did he hate about math?  “Arguments.”

“What kind of math have you been doing,” I wondered in my head.  “Sounds fantastic!”  But I didn’t say that aloud.  Instead, I acknowledged that there definitely can be arguments in math.  And I confirmed for them that the “none-left-over-means-even” definition assumes that the two parts are integers or whole numbers.  I moved the students to another whiteboard for a new activity.  They seemed happy for me to be done messing with them.



Rachel turned her back on the students and the board.  “Let’s see if Rachel can correctly predict whether the sum of some numbers is even or odd.  Each of you come up and write an odd number on the board.”  They did.  “How many odd numbers do we have?”  Five.  “Rachel, we have five odd numbers on the board.  Do you think the sum will be even or odd?”  Odd.  “Turn around and watch while we figure out if you were right.”  I expected the kids to balk at this part of the activity, but they embraced it cooperatively, added the numbers two-by-two, and voila, Rachel was right!

S, L, and B then took turns with their backs turned to make the prediction.  M preferred to contribute odd numbers.  Four rounds of this game took a while, but the kids never tired of it.  They also didn’t shy away from large numbers once they realized the task of adding the numbers was on them.  They chose numbers in the millions and billions.  They had fun writing them on the board themselves.  (See photos of their boardwork.)  Each student’s parity (even vs. odd) prediction was correct.  “How do you do it?” I asked.

“Something about the number of things on the list,” said B.  “If that number is even or odd you can tell.”  M and S agreed.  L, however, contended that a magical frog was whispering the answers in people’s ears.  He placed a small stuffed frog on the table.  I wrote both conjectures on the board, and labeled them “conjectures.”

“That word sounds like the name of a sickness where you can’t go to the bathroom.”

“It’s a different word,” I explained.  “Have any of you ever heard the word conjecture before?”  No one had, which surprised me since all of these kids have been in math circles with me before.  Was I not using that word with the younger ones?  “I know how we can test which conjecture is accurate.  Let’s do our next activity outside, and leave the frog in the building, out of earshot.  The frog won’t be able to help.  Then we’ll know if the solutions are coming from math or a magical frog.”  We moved outside.



We played another parity game outside – Treasure Hunting, which involves crossing an imaginary river a certain number of times.  The students made predictions, some accurate and some not-so-accurate, then used a jumprope as the river and tested their conjectures.  Giggling, they attributed the accurate predictions to the magical frog.  We’ll continue this game next week.



We played one final game of NIM – this time against me.  They beat me.  “We won!  We’re smarter!” they cried with glee.  They want to play it more to develop a reliable strategy.  I told them we’ll do it every week.  I also told them that they can play it online at  This particular game does not start at the easiest level.

Almost all of the credit for the activities in this class go to  Amanda Sereveny, Julia Brodsky, Emily McCullough, Maria Droujkova, and Ashley Ahlin.  Thanks to all of you!

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