MHB How Do You Determine the Number of Toothpicks in a Stack of Squares?

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To determine the total number of toothpicks needed for a stack of squares, a quadratic function can be used based on the sequence of toothpick counts: 4, 10, 18, and 28. The first differences (6, 8, 10) indicate a constant second difference of 2, confirming a quadratic relationship. The function can be expressed as T(n) = k_1n^2 + k_2n + k_3, where the parameters k_i are derived from the given values. By solving the resulting linear system, it is found that T(n) = n^2 + 3n, which simplifies to n(n + 3). This provides a clear formula for calculating the toothpicks needed for any stack height n.
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I am doing a problem in math which I can't seem to solve.The problem is apart of my discrete functions unit it grade 11 math. The question is: Toothpicks are used to make a sequence of stacked squares as shown. Determine a rule for calculating the total number of toothpicks needed for a stack of squares that is n high. It also gave a diagram with 4 terms and the number of toothpicks in each term was 4,10,18,28. I know that the first differences are 6,8 and 10 which means the second differences share a constant value of 2. Does this mean that I need to make a quadratic equation to find the answer or something else? If so, how would I? By the way, this problem is a thinking or extension problem.
 
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Your observations are correct...we can represent the total $T$ number of toothpicks for a stack $n$ levels high with the function:

$$T(n)=k_1n^2+k_2n+k_3$$

So, what you want to do is use the given values to determine the values of the parameters $k_i$.

$$T(1)=k_1+k_2+k_3=4$$

$$T(2)=4k_1+2k_2+k_3=10$$

$$T(3)=9k_1+3k_2+k_3=18$$

So, you have a linear 3X3 system to solve.

By the way, I am going to delete the duplicate of this thread posted in our "Chat Room" forum. :D
 
We could simplify matters by observing $T(0)=0$ and since then $k_3=0$ reduce the system to:

$$k_1+k_2=4$$

$$2k_1+k_2=5$$

Subtracting the former from the latter, we obtain:

$$k_1=1\implies k_2=3$$

And so we have:

$$T(n)=n^2+3n=n(n+3)$$
 
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