How to Calculate Total Toothpicks and Squares in Any Figure of the Sequence?

In summary, the conversation discusses finding the number of toothpicks and squares in a figure sequence and using patterns to create a formula for any figure number without listing them all out. A suggestion is made to count the toothpicks and see a pattern, and a table is created to show the number of toothpicks and squares for each figure. The problem is not being able to find a formula for any figure number. The expert then gives hints to use an arithmetic progression formula with a first term of 4 and a common difference of 2.
  • #1
marsman
2
0
Please help! I need to be able to find the number of total toothpicks used in any given figure in this sequence, in addition to total squares in any figure, using the figure number.View attachment 1592
The only way I can figure this out is by listing them all out one by one, which is TERRIBLE :(
Thank you so much in advance!
 

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  • #2
marsman said:
Please help! I need to be able to find the number of total toothpicks used in any given figure in this sequence, in addition to total squares in any figure, using the figure number.View attachment 1592
The only way I can figure this out is by listing them all out one by one, which is TERRIBLE :(
Thank you so much in advance!
Suggestion: In the six figures, count how many toothpicks you have to add to each figure in order to get the next one. Can you see a pattern there?
 
  • #3
Opalg said:
Suggestion: In the six figures, count how many toothpicks you have to add to each figure in order to get the next one. Can you see a pattern there?

Yeah I'm up to my neck in patterns; I made a table and everything.
Figure Number12345
Number of Toothpicks410182840
Increase in Toothpicks+4+6+8+10+12
Number of Squares1361015
Increase in Squares+1+2+3+4+5
My problem is that I can't get from the patterns to creating a formula that would work for any figure number without listing them all out consecutively
 
  • #4
I see you have found that the difference in the number of toothpicks per iteration (where you state the increase in toothpicks) are the sequence of even numbers, beginning with 4. What is the second difference, that is, the difference of the differences. How much does the increase increase each time?
 
  • #5
marsman said:
Yeah I'm up to my neck in patterns; I made a table and everything.
Figure Number12345
Number of Toothpicks410182840
Increase in Toothpicks+4+6+8+10+12
Number of Squares1361015
Increase in Squares+1+2+3+4+5
My problem is that I can't get from the patterns to creating a formula that would work for any figure number without listing them all out consecutively
Let $x_n$ be the number of toothpicks in Fig. $n$. Your table shows that $x_1=4$, $x_2=4+6$, $x_3=4+6+8$, $x_4=4+6+8+10$, $\ldots$. So what is the formula for $x_n$?

[sp]Hints: arithmetic progression, $n$ terms, first term $4$, common difference $2$.[/sp]
 

FAQ: How to Calculate Total Toothpicks and Squares in Any Figure of the Sequence?

What is the Toothpick Squares Sequence?

The Toothpick Squares Sequence is a mathematical sequence that involves using toothpicks to create a pattern of squares. It was first discovered by mathematician Marjorie Rice in 1977.

How do you create the Toothpick Squares Sequence?

To create the Toothpick Squares Sequence, start with a single toothpick. Place two toothpicks perpendicular to the first one, forming a "T" shape. Continue this pattern, adding two toothpicks each time, until you have a square. Then, add one toothpick to each side of the square to create a larger square. Repeat this process to create a sequence of larger and larger squares.

What is the significance of the Toothpick Squares Sequence?

The Toothpick Squares Sequence is significant because it is an example of a self-similar, or recursively defined, pattern. This means that the same pattern is repeated at different scales, creating an infinite sequence. It also has applications in geometry and number theory.

What is the relationship between the number of toothpicks and the number of squares in the sequence?

The number of toothpicks in each square is equal to the number of toothpicks in the previous square plus four. This means that the total number of toothpicks in the sequence can be calculated using the formula n^2 + n + 1, where n is the number of squares.

Are there any variations to the Toothpick Squares Sequence?

Yes, there are several variations to the Toothpick Squares Sequence, such as starting with different initial shapes (such as a triangle or pentagon) or using different types of toothpicks (such as colored or differently sized toothpicks). These variations can lead to different patterns and sequences.

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