Download Toothpick problems

The Window Factory
Although you are an eighth grader, you actually have become an engineer in your great
uncle’s window architectural firm! This firm specializes in making grids for windows. It is
clear, however, that things need to be modernized a bit. Instead of making a prototype
for every possible size of grid, the firm needs formulas so that the right number of
wooden rods and connectors can be ordered immediately. You can wow your great
uncle by figuring out the formulas.
Here is what each grid needs:
1.
2.
3.
4.
Wood segments (a.k.a. toothpicks)
“L” connectors to connect two wooden rods at right angles
“T” connectors to connect three wooden rods at right angles
“X” connectors to connect four wooden rods at right angles
Example:
a 3-by-3 window
How many wooden rods would be needed for any size square window?
How many of each type of connectors would be needed for any size square grid?
Write a memo to your great uncle showing the research you have done. Your formulas
must convince him that they are correct (if they are wrong, the firm will lose a lot of
money!). Use sketches and explanations to show clearly why your formulas make sense
with the geometry of the grids. Use graphs to support this reasoning.
Further questions to consider:
1. What is the largest size window you can make with 125 X-connectors and 49 Tconnectors?
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Window Factory
2. What if the window is not a square? What if it were a non-square rectangle?
Try windows that have one row and varying numbers of columns. Find general formulas
for numbers of wooden rods and each of the types of connectors.
Try windows that have two rows and varying numbers of columns. Find general
formulas for numbers of wooden rods and each of the types of connectors.
Try windows that have three rows and varying numbers of columns. Find general
formulas for numbers of wooden rods and each of the types of connectors.
Now try to generalize for m rows and n columns.
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Window Factory
3. Suppose the windows were made in equilateral triangular shapes as shown below.
Find how many wooden rods are needed for an n-sided (n toothpicks on each side)
triangle?
How would you define the connectors in this case? Define all the different kinds of
connectors needed and find general formulas for the numbers of each type of
connector needed for an n-sided triangular window.
A 3-sided triangular window
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Window Factory