# Problem Of The Week #466 May 3rd 2021

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#### anemone

##### MHB POTW Director
Staff member
Here is this week's POTW:

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A person is working on a jigsaw puzzle that contains 1000 pieces. It is found that it takes 3 minutes to put the first two pieces together and that when $x$ pieces have been connected it takes $\dfrac{3(1000-x)}{1000+x}$ minutes to connect the next piece. Determine an accurate estimate of time (in hours) it takes to complete the puzzle.

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#### anemone

##### MHB POTW Director
Staff member
No one answered last week's POTW. However, you can find the model solution as follows:
The exact number $N$ of minutes to complete the puzzle is $\displaystyle \sum_{x=0}^{999}\dfrac{3(1000-x)}{1000+x}$. Since $\dfrac{3(1000-x)}{1000+x}$ is a non-negative monotonic decreasing function for $0\le x \le 1000$, we see that

$N-3\le \int_0^{1000}\left(-3+\dfrac{6000}{1000+x}\right) dx\le N$

Therefore $\dfrac{N}{60}\approx 50(2\ln 2-1)$. Using $\ln 2\approx 0.69$, we conclude that it takes approximately 19 hours to complete the puzzle.

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