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## Question

This one my sir asked me today and I want to share it with all of you

It says that; what is the maximum number of pieces which could be cut out from a circular dish (PIZZA) using only 12 strokes(cuts)

Even I am struggling .TRY THIS PLEASE

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56?

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2^12 = 4096

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n*(n+1)/2 + 1

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n*(n+1)/2 + 1

???

Formula for what? Not only that, what good is a formula where you don't describe anything? What is 'n' in your formula? The number of cuts?

the most new pieces you could with a single cut would be equal to the number of cut you are making. As an example if you start with a pizza cut if half and you add a second cut to it the most number of pieces you can add would be 2. For your 3rd cut you can add 3 more pieces, for your 4th cut you can add 4 more pieces... etc.

Therefore my answer is:

```
# of cuts | Max Number of pieces

1         | 2

2         | 4

3         | 7

4         | 11

5         | 16

6         | 22

7         | 29

8         | 37

9         | 46

10        | 56

11        | 67

12        | 79

```

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can you all please upload pictures so that it can be clear

and I got only 56

THANKS TO ALL OF YOU FOR HELPING

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and witzar according to you is the answer 52 if not so please explain the formula

thanks in Advance

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Cut the pizza in half, then superpose the halves and then cut again in half. Continue this way and you nave 2^n pieces at the end of n cuts. So for 12 cuts, you have 2^12 = 4096.

By cutting in a grid and putting the pizza into it, you have (1+(n/2))^2 pieces of the pizza. Therefore in this case it is 49. However, the pieces are not identical in shape.

Edited by Adhish Majumdar
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Formula for what? ... What is 'n' in your formula? The number of cuts?

I think it's pretty clear that the formula is to calculate the maximum number of pieces one can get from n straight cuts.

Not only that, what good is a formula where you don't describe anything?

Sometimes people don't have enough time to provide full detailed description of their solution. I'm one of those people. That doesn't make their answer incorrect or not valuable. Given the general formula provided by witzar anybody can calculate the max number of pieces resulting from any number of straight cuts.

and witzar according to you is the answer 52 if not so please explain the formula

thanks in Advance

12*13/2+1 = 79

The formula is based on the fact that with every N-th cut you will be able to cut at most N pieces (can anybody prove this?)

So, if you make a correct cut you will increase the total by N with every N-th cut. So, the formula is simply a sum of all numbers from 1 to N plus 1 additional piece that you get with the first cut.

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...

Thanks, k-man. You have explained it all perfectly.

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I think it's pretty clear that the formula is to calculate the maximum number of pieces one can get from n straight cuts.

Sometimes people don't have enough time to provide full detailed description of their solution. I'm one of those people. That doesn't make their answer incorrect or not valuable. Given the general formula provided by witzar anybody can calculate the max number of pieces resulting from any number of straight cuts.

12*13/2+1 = 79

The formula is based on the fact that with every N-th cut you will be able to cut at most N pieces (can anybody prove this?)

So, if you make a correct cut you will increase the total by N with every N-th cut. So, the formula is simply a sum of all numbers from 1 to N plus 1 additional piece that you get with the first cut.

You "don't have enough time" to simply add "Where n is the number of cuts?" Wow you must be one busy person. Regardless we came up with the same solution. When I found a pattern I assumed their had to be a formula; I just couldn't figure out what it was other than the pseudo formula I used of adding the number of cut you are making.

While I don't have a proof that says you can only get N pieces from each cut; it seems logical to me. Since you're dealing with straight cuts each new cut can only cut through each previous cut once.

Shivam it's too difficult to create a gif of this with 12 slices. Besides, I agree with Witzar; who has the time for that. With every new cut make sure you cut through each previous cut. It gets hard to demonstrate after about 8 cuts.

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The formula is based on the fact that with every N-th cut you will be able to cut at most N pieces (can anybody prove this?)

You start and finish the n-th cut at some two points at the edge of the pizza.

Ont it's way this cut crosses each of n-1 previous cuts.

When the n-th cut crosses first previous cut, a new piece appears. When it crosses second cut we have second piece, and so on.

When we cross last previous cut we have (n-1) new pieces, and finally we reach the edge of the pizza which produces one more piece.

This makes total of n new pieces with n-th cut.

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You "don't have enough time" to simply add "Where n is the number of cuts?" Wow you must be one busy person.

It is not that I'm a busy person, it just didn't occur to me that there could be any doubt about what n stands for.

Anyway, my apologies to everyone confused.

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There's a few ways to approach this.

Linear radial strokes have already been addressed, as well as the grid approach, and I didn't even think of super positioning, but that's here too. So that just leaves circular cuts.

This is assuming each stroke does not need to be linear and that each piece does not need to be identical. Basically, I thought why not spare a few strokes to make smaller circles within the whole.

Basically, I cut 7 linear strokes to create 14 equal pieces. Then I cut 5 smaller circles (all sharing the same center) into the shape, effectively multiplying the number of pieces by 6 (since when dividing an object, the outcome is always divisions+1).

Formula: (r*2)*(c+1)

r = linear radial strokes

c = circular strokes

With this technique, I made 84 pieces.

Rough Visual Aid (not 100% accurate)

This got me thinking further and I figured why make multiple cuts when you can simply cut one continuous spiral and afford the rest into linear radial strokes.

So here I reduced the circular cuts to 1, but instead of creating one closed circle, I cut one continuous spiral from the edge to the center and used the remaining 11 strokes for linear radial cuts.

Formula: (r*2)*(R+1)

r = linear radial strokes

R = full revolutions within the spiral

With this technique (using 7 full revolutions), I made 176 pieces; but given a tight enough spiral, this number could be significantly larger; theoretically infinite.

Rough Visual Aid (again, not 100% accurate)

Though, after reading the answers so far, I conclude that if super positioning is allowed, then the most slices may be attained by a combination of that plus linear and circular strokes; but I haven't done the math for that possibility yet.

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@Vagabond

I love how you colored it in. I wish you had animated it to spin, though.

That you could cut out a bullseye pattern. That would give you a lot of slices, but I think the spiral is better because that's only one long cut.

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1st cut - +1

2nd cut - Either it doesn't cross the first cut (+1) or it does (+2)

3rd cut - Either doesn't cross a cut (+1), crosses one cut or center (+2) or crosses the two cuts (+3)

To get the +N the next cut has to cross all the other cuts but not at an intersection with another cut...not sure if this is possible though.

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Since all the lines are straight at most a line can only cross another line once.

Each line will add at least one more area and another area for each line it crosses that doesn't cross at an intersection with a 3rd line.

As for drawing, start with two lines dividing the circle into 4 equal parts. Each line after shift along an axis and tilt a small bit. Repeat shifting and increasing the tilt from the previous line.

The areas will be really small but it will work as long as the shift and tilt are small enough so that the nth line stays within the circle after crossing the lines.

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With just putting in random straight lines, I was able to get 62 pieces. I'm sure there is a configuration where you can get more than that, but you can at least do 62.

It's a really crude picture, but it was the best I could do. The black numbers are the pieces. The green numbers are the 12 lines.

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It could be infinite, because it never says that the cuts must be straight. Also:

You could have a lot more than 56, because you could cut it horizontally. You would double the number of slices if you made that the final cut, and it would be really high.

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Ok I got the formula But I am still unable to draw it on paper

PLEASE HELP

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