Left-Sided Rhombuses in a Larger RhombusDate: 05/22/2000 at 14:34:55 From: Stuart Wright Subject: Formulas for individual rhombuses inside a larger rhombus This question is similar to one you have previously answered, though different. My math coursework is to work out a formula for the number of individual rhombuses inside a larger rhombus, say 3x3 or 4x4. These formulas must be for number of right-sided rhombuses, left-sided rhombuses, and vertical rhombuses (diamonds.) I already have the table and formulas for right-sided; let me show it to you. Size of rhombus 1x1 2x2 3x3 4x4 5x5 6x6 1x1 1 0 0 0 0 0 2x2 4 1 0 0 0 0 3x3 9 4 1 0 0 0 4x4 16 9 4 1 0 0 5x5 25 16 9 4 1 0 6x6 36 25 16 9 4 1 Formula n^2 (n-1)^2 (n-2)^2 (n-3)^2 (n-4)^2 (n-5)^2 If you imagine a rhombus 3x3, it has a potential number of 26 rhombuses inside: 21 individual 1x1 rhombuses (including right + left + vertical sided), 4 2x2 rhombuses and 1 3x3 rhombus. However, I require a table and formula for left-sided and vertical rhombuses like the one above, and I have no idea what it is. If you could provide me with that information I would be very grateful. Also, we have to explain how the formula works. If you can, could you give me an idea of that? Thanks in advance. Date: 05/22/2000 at 17:13:01 From: Doctor Peterson Subject: Re: Formulas for individual rhombuses inside a larger rhombus Hi, Stuart. I'd rather tell you how the formula works, and let you find it. So let's start by looking at the formula you already have. I presume your figure is something like this: +---+---+---+---+ / \ / \ / \ / \ / +---+---+---+---+ / \ / \ / \ / \ / +---+---+---+---+ N = 4 / \ / \ / \ / \ / +---+---+---+---+ / \ / \ / \ / \ / +---+---+---+---+ What you counted are shapes like +---+ +---+---+ / / / / +---+ and + + K = 1 / / +---+---+ K = 2 How did you count them? I would do it by seeing how many places there are to put, say, the upper left corner. For K = 1, there are N positions left-to-right, and N positions top-to-bottom, so there are N*N = N^2 in all. If K (the size of the small rhombus) is increased by 1, that takes away one possible position in each direction; so for any K there are (N+1-K)^2 of them. That gives the numbers you gave. For example, with K = 2, the (4+1-2)^2 = 3^2 = 9 points marked with "o" are possible locations of the upper left corner: o---o---o---+---+ / \ / \ / \ / \ / o---o---o---+---+ +---+---+ / \ / \ / \ / \ / / / o---o---o---+---+ + + / \ / \ / \ / \ / / / +---+---+---+---+ +---+---+ / \ / \ / \ / \ / K=2 +---+---+---+---+ Do the same for each of the other orientations of the rhombus. For example, here are the places where the top corner of the "vertical" rhombus with K = 1 might be: o---o---o---o---+ / \ / \ / \ / \ / o---o---o---o---+ + / \ / \ / \ / \ / / \ o---o---o---o---+ + + / \ / \ / \ / \ / \ / +---+---+---+---+ + / \ / \ / \ / \ / K=1 +---+---+---+---+ The formula will be similar to the other, but will be a rectangular number rather than a square. Also, you lose not one but two possible rows when you add one to K, so there will be a multiplication by 2 in the formula. Let me know if you need more help; I'm hoping this will give you what you need to be able to find the other formulas yourself. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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