who: J1
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}