where: at Kuu, our regular lunch destination at the mall

when: while waiting for ice cream

what did we use: chopsticks

As we enter this snippet of conversation, J1 was describing/acting out some martial fantasy scene with different elements battling and weapons getting destroyed.

J1: What if I broke his sword into 11 pieces?

J0: 11 equal pieces?

J1: yes, 11ths. Then he would cry 11 times.

J0: I wonder, is it easy to cut something into 11 equal pieces?

J1: no, hard to get them all the same size.

J0: What is easier, cutting into halves or thirds?

J1: (thinks for a bit) cutting into half is easy. (He then makes a swipping motion with his chopstick and a blade-swooshing sounds.) If you cut it into thirds, you do chop (one smaller slice), chop (another slice about 120 degrees of the previous one), and

**then you really need to put your back into it**(as he makes a third slice toward himself).

J0: (laughing) What did you say?

J1: (laughing, then repeats the last cutting motion)

**then you really need to put your back into it.**

J0: Wow, you were dividing a round cake. I was just thinking of sticks. How many cuts do you need for those?

We proceeded to have some discussion of how many cuts were needed, some contemplation of why a it takes n radius cuts to divide the cake into n pieces (for n>2), and a hypothesis about why halves are the easiest to divide (because you only have to compare two pieces and make one cut).

One further comment was worth flagging, related to my n>2 provision above:

J1: hmm, thirds take 3 cuts, but halves only take one cut?

As this was in the middle of something else he was describing, I didn't follow that branch of the conversation, but may return to it later.

*

**Apology*** sorry I didn't include any pictures on this post. I'll see if we can draw a picture of the enemy with a sword broken into 11ths.

Should be n greater than or equal to two. You need two radius cuts to cut a circle in half. The "one cut" you refer to is a,diameter which isn't allowed if you specify radius cuts. If you are just talking cuts then you need n for odd number, but n/2 for even as you can do it with diameters, not radii.

ReplyDeleteIndeed. This was a loose conversation (warm-up) without any pictures or deeper investigation, so some of the claims weren't correct (or consistent). Part of the follow-up will involve the question of "what is a cut?" and I expect we will reconcile/correct some of the earlier hypotheses.

DeleteFor now, I thought it was interesting that J1's immediate intuition was to use radius cuts for 3 pieces, to count those each as cuts, and to generalize this for larger numbers of pieces, but his sense of a single "cut" is a line through the object.

Sticking to a circle being shared, I have come up with 5 possible definitions of a "cut," and am curious to see if the discussion unearths versions I haven't considered. For reference, here are my 5:

- radius cuts

- diameter cuts

- straight lines (don't have to pass through the centre, but do run from edge to edge)

- straight line segments of common fixed length

- straight lines that run from {edge or pre-existing cut} to {edge or pre-existing cut}