Sticky Gum Problem POW

From ThePlaz.com

Revision as of 01:38, 22 June 2007 by ThePlaz (Talk | contribs)

Jump to: navigation, search

IAG 1H POW # 4: A Sticky Gum Problem

Word.PNGA Microsoft Word version of this work is available here: Image:A_Sticky_Gum_Problem.doc

Pdf.jpgA PDF version of this work is available here: Image:A_Sticky_Gum_Problem.pdf


Contents

Problem Statement

Not necessary to do.

Process and answers to problems 1,2 & 3

1. (2 colors, 2 people) Ms. Hernandez can only spend 3 cents, because on her first cent, she can get a white or a red gumball. On her second turn, she can also only get a red or white gumball. Now she can only have 4 combinations: Red and White Gumballs; Red and Red Gumballs; White and White Gumballs; White and Red Gumballs. With 2 of these possible combinations, she already has her goal of having 2 of the same color gumballs. On her third try, she gets another red or white gumball. Whatever the color, she already has one of them, which makes 2 of the same color.

Sticky Gum Problem Diagram 1.png

2. (3 colors, 2 people) Ms. Hernandez now finds a machine that has 3 colors in it. The most that she will need to spend to get 2 of the same color is 4 cents. To find all of the possible strategies, look at the chart:

Sticky Gum Problem Diagram 2.png

3. (3 colors, 3 people) It will take 7 cents to get 3 of the same color, as there are 108 possible combinations. Here is a chart showing the 1st third of them (if the first color is red)

Sticky Gum Problem Diagram 3.png

I will now make a chart showing my findings so far.

Colors Kids Max Spend Combos
2 2 3 cents 6
3 2 4 cents 33
3 3 7 cents 108

Now I will make up some problems to help fill in the chart some more.

4. (2 colors, 3 kids) This chart shows the first half of needing 3 of the same color, with only 2 colors. It will take 5 cents and there are 16 combinations.

Sticky Gum Problem Diagram 4.png

5. (2 colors, 4 kids) This shows the first half of getting 4 of 1 color and having 2 colors. You need 7 cents, and there are 68 combos.

Sticky Gum Problem Diagram 5.png

6. (4 colors, 2 kids) This shows the first quarter of the chart, when you need 2 of the same, and there are 4 colors. You need 5 turns to get 4 of the same, and there are 200 combinations.

Sticky Gum Problem Diagram 6.png

Solution (Ultimate Goal)

Let me make another chart. I have included 1 color and 1 kid for comparison.

Colors Kids Max Spend Combos
1 1 1 1
1 2 2 1
2 1 1 2
2 2 3 cents 6
2 3 5 16
2 4 7 68
3 2 4 cents 33
3 3 7 cents 108
4 2 5 200


Overall, I have found that number of colors is ultimately responsible for combinations, but the number of kids is ultimately responsible for the maximum, you spend. Here is something interesting:

Colors Kids Max Spend
2 2 3
3 4
4 5

When you have 2 kids max spend is equal to number of colors plus 1. What about having 3 kids:


Colors Kids Max Spend
2 3 5
3 4

This chart shows so far that when you add a color, the max that you spend, goes down.

When you start adding a color, the number of cents goes down; when you then keep adding people, the number of cents goes up.

I have also found that this works: [(# of colors) * (# of kids)] - [(# of colors) – 1]

Extension

Not necessary to do.

Evaluation

Not necessary to do.