How did the Lazy Caterer's Sequence come about?
There is an interesting pizza (or any round food) formula, which has nothing to do with the toppings. Instead, it involves cutting the pizza up into separate bits (but they are not equal pieces). This makes a nice puzzle that you can try for yourself!
Imagine that you've got a huge round pizza and a very long straight knife (please be careful)! Before you touch the pizza, you have one very large piece. If we cut the pizza once (exactly in half), both pieces will be the same size. If we also cut the pizza again (where it passes through the centre, we will get 6 equal pieces and then 8 pieces and so on.
Let's mix things up! We want to cut up a new pizza, but we don't care about the size or the shape of the pieces. Instead, we want to try and find the biggest number of pieces you can make with each cut. For example, if we cut the pizza once, the maximum number of pieces is still 2, and if we cut it twice, the maximum number of pieces is still 4 (as can be seen below).
Imagine that you've got a huge round pizza and a very long straight knife (please be careful)! Before you touch the pizza, you have one very large piece. If we cut the pizza once (exactly in half), both pieces will be the same size. If we also cut the pizza again (where it passes through the centre, we will get 6 equal pieces and then 8 pieces and so on.
Let's mix things up! We want to cut up a new pizza, but we don't care about the size or the shape of the pieces. Instead, we want to try and find the biggest number of pieces you can make with each cut. For example, if we cut the pizza once, the maximum number of pieces is still 2, and if we cut it twice, the maximum number of pieces is still 4 (as can be seen below).
You will get a chance to cut up this new pizza in the activities below. For example, what is the biggest number of pieces you can make with five or six cuts? Are there any patterns?
A formula to find the maximum pieces of pizza does exist! Can you figure out what it is?
But the next time you invite friends over, don't think, "Hey, I only need 1 pizza! Surely they won't mind if their pieces are different sizes!" Please note that if you do manage to do this, the friends who receive the smaller pieces will be quite annoyed.
A formula to find the maximum pieces of pizza does exist! Can you figure out what it is?
But the next time you invite friends over, don't think, "Hey, I only need 1 pizza! Surely they won't mind if their pieces are different sizes!" Please note that if you do manage to do this, the friends who receive the smaller pieces will be quite annoyed.
Time to Explore
1. Draw a big circle that represents your pizza, pancake or any round food that you desire.
Find the maximum number of pieces for:
(i) 3 cuts
(ii) 4 cuts
(iii) 5 cuts
(iv) 6 cuts
(v) 7 cuts (this is a challenge)
2. Can you find any patterns? What are the differences in the number of pieces of pizza as the number of cuts increase?
3. What is the best strategy for cutting the pizza to ensure that you do get the maximum number of pieces?
4. Try and devise a formula so that we can find the maximum number of pieces for any number of cuts.
5. Does your formula work for 1-7 cuts, as found in Question 1? What is the biggest number of pieces I can make if I cut the pizza 10 times? How about 50 times?
Find the maximum number of pieces for:
(i) 3 cuts
(ii) 4 cuts
(iii) 5 cuts
(iv) 6 cuts
(v) 7 cuts (this is a challenge)
2. Can you find any patterns? What are the differences in the number of pieces of pizza as the number of cuts increase?
3. What is the best strategy for cutting the pizza to ensure that you do get the maximum number of pieces?
4. Try and devise a formula so that we can find the maximum number of pieces for any number of cuts.
5. Does your formula work for 1-7 cuts, as found in Question 1? What is the biggest number of pieces I can make if I cut the pizza 10 times? How about 50 times?
Challenge
6. Let's see what happens if we do some chopping in three dimensions.
Imagine that you have a block of cheese. As can be seen in the picture on the right, we can cut the block of cheese horizontally as well as vertically. If we do cut the block of cheese three times, we can get a maximum of 8 pieces.
(i) What is the maximum number of pieces of cheese you can get with 4 cuts? How about 5 or 6 cuts?
(ii) Devise a formula so you can find the maximum number of pieces of cheese possible with any number of cuts. Using this formula, how many pieces can I get with 7 cuts?
(iii) What are the differences in the number of pieces of cheese as the number of cuts increase? What do you notice?
Imagine that you have a block of cheese. As can be seen in the picture on the right, we can cut the block of cheese horizontally as well as vertically. If we do cut the block of cheese three times, we can get a maximum of 8 pieces.
(i) What is the maximum number of pieces of cheese you can get with 4 cuts? How about 5 or 6 cuts?
(ii) Devise a formula so you can find the maximum number of pieces of cheese possible with any number of cuts. Using this formula, how many pieces can I get with 7 cuts?
(iii) What are the differences in the number of pieces of cheese as the number of cuts increase? What do you notice?