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Left-Sided Rhombuses in a Larger Rhombus

Date: 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?

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/

Associated Topics:
High School Geometry
High School Puzzles
High School Triangles and Other Polygons
Middle School Geometry
Middle School Puzzles
Middle School Triangles and Other Polygons

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