Cutting a SquareDate: 05/05/99 at 10:50:56 From: Nickolas Joubert Subject: Maths investigation - "cutting cake" I have been given this math investigation to do. I am completely stuck and I wondered if you could help me. Given a square that is any size, I have to "cut" the square ten times and see how many pieces I can obtain. I have to obtain as many pieces as possible. For example, a square can be cut into two with one straight cut. That same square can be cut into four pieces with two straight cuts. However, I can obtain seven pieces with three straight cuts. I hope you see what I mean. I have worked out a pattern and have found out how many pieces can be obtained from ten cuts, which is fifty-six. I understand it up to there, but now I need to work out a formula. I need the formula so that I don't need to go through a lot of work to out to find out how many pieces can be obtained from a much higher number of cuts, for example, seventy.) I would be very grateful if you could help me. Thanks very much, Nickolas Joubert Date: 06/11/99 at 22:30:00 From: Doctor Jeremiah Subject: Re: Maths investigation - "cutting cake" Hi Nickolas: If your pattern is 2, 4, 7, 11, 16, 22, 29, 37, 46, 56... then we should be able to find a formula. I don't know for sure whether this pattern is right or not. It does look like it though. I am sure that 2, 4, 7, 11, 16 is right (I checked them myself). Lets assume that the pattern holds for bigger numbers. To create a formula for this we need to figure out what we are doing to one number to get the next number. This is easy; we are always adding a value one bigger than the last number we added. 2 = 1 + (1) 4 = 1 + (1 + 2) 7 = 1 + (1 + 2 + 3) 11 = 1 + (1 + 2 + 3 + 4) 16 = 1 + (1 + 2 + 3 + 4 + 5) 22 = 1 + (1 + 2 + 3 + 4 + 5 + 6) 29 = 1 + (1 + 2 + 3 + 4 + 5 + 6 + 7) 37 = 1 + (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8) 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ... is a summation. The formula for this summation just happens to be n(n+1)/2, where n is the largest number you are adding. 2 = 1 + n(n+1)/2 <== n = 1 4 = 1 + n(n+1)/2 <== n = 2 7 = 1 + n(n+1)/2 <== n = 3 11 = 1 + n(n+1)/2 <== n = 4 16 = 1 + n(n+1)/2 <== n = 5 22 = 1 + n(n+1)/2 <== n = 6 29 = 1 + n(n+1)/2 <== n = 7 37 = 1 + n(n+1)/2 <== n = 8 So n is the number of cuts, and 70 cuts would make: 1 + 70(70+1)/2 = 1 + 2485 = 2486 pieces If you want to know where the n(n+1)/2 comes from let me know. - Doctor Jeremiah, The Math Forum http://mathforum.org/dr.math/ |
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