Toothpick Squares Problem
Toothpick Square Write Up:
The toothpick square problem was a problem were there is a set of squares made out of tooth picks. The first square is made out of four toothpicks, the second one is made out of twelve toothpicks and is comprised of four inner squares, and the third one is comprised of 9 squares and is made of 24 toothpicks. The pattern increases by one square up and one square over each time. We had to figure out a way to find how many toothpicks were in the 100th and Nth square, without making the square and counting.
To solve this problem I started by looking at the number of squares that were with in the squares. By doing this all I noticed that the number of squares was just the number squares squared but this didn't help me with the problem because the problem was asking for the amount of toothpicks in the square, not the amount of squares with in the square. I started looking at the amount of toothpicks that changed between each set, when I counted the amount I got was, the 1st one had 4, the 2nd one had 12, the 3rd one had 24, the 4th one had 40, the 5th one had 60, and the 6th one had 84. Looking at this I was trying to figure out whether or not they had something in common, or related to each other. I noticed that they were all divisible by 4, and the 1st was 4 multiplied by 1 and 4, then 4 multiplied by 3, and 4 multiplied by 6. This was a pattern because it went 1, 3, 6. So 1st you add 2, then you add 3. But after those 3 numbers it did not work. So I went on to looking at another way of solving. I kept looking for certain patterns. While I was doing this I was talking to George about how he was solving the problem. He was coming up with these methods that didn’t work for all of them, but worked for a few of them, similar to what what I was coming up with. While I was looking of the difference in the numbers between 4, 12, 24, 40 and 84 I noticed a pattern. The difference between 4 and 12 equals 8, when you add 4 to it, it equals 12, when you add 12 to 12 equals 24, then you take the difference between 12 and 24 which is 12 and and you add 4, which equals 16, then you add 16 to 24 which equals 40, which is also the difference between 24 and 40. After you do that you continue the process of taking the difference in adding 4. If you take 20, the difference between 40 and 60, add 4 which was 24 and 24 to 60 it equals 84. Then you would add 24 with 4 and then add that to 84 and he would give you the next number in the sequence. Therefore to find the 100th or and pattern you just need to do the process of adding them. The 1st 13 are 4, 12, 24, 40, 60, 84, 112, 144, 180, 220, 264, 312, 364. I got this answer because I used the process of adding the difference between each one with the number 4. Unfortunately I did not find any equation they could instantly tell me the answer pattern so to solve the nth pattern.
To find 100 pattern I used a pattern that I found where you add 4 to the difference between the last digit, and I came up with the number 20, 228 toothpicks in the 100 square in the pattern. To find the nth pattern you have to go through each number of squares and use the method that I found.
I have learned to look for patterns and equations while doing this problem and while looking at all the in and out tables. If I were to give myself a grade on this problem I would give myself a 10/10 because I found a way to solve the problem very early on, consulted with other students on ow they solved their problems, and helped some students with solving it. During this problem I used supposition because looking at this problem and the other tables there can be different equations under different circumstances to explain the in converting to the out. For my solution I found a complex ay of doing it, but George seemed to find a less complex version of solving it using equations, there fore we did it differently under different circumstances.