Here is another sum problem, this time from the 2021 Math Calendar.

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As before, recall that all the answers are integer days of the month. And the solution employs a technique familiar to these pages.

See the Winter Sum

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Here is another sum problem, this time from the 2021 Math Calendar.

________________

As before, recall that all the answers are integer days of the month. And the solution employs a technique familiar to these pages.

See the Winter Sum

Here is a challenging problem from the 2021 Math Calendar.

“Find the remainder from dividing the polynomial

*x*^{20} + *x*^{15} + *x*^{10} + *x*^{5} + *x* + 1

by the polynomial

*x*^{4} + *x*^{3} + *x*^{2} + *x* + 1”

Recall that all the answers are integer days of the month.

See the Remainder Problem

Here is an intriguing problem from the 2021 Math Calendar.

“If the smaller circle of diameter 7 rotates without slipping within the larger circle, what is the length of the path of P?”

The problem did not state clearly how far the smaller circle should rotate. Its answer implied it should complete just one full (360°) rotation within the larger circle.

Recall that all the answers are integer days of the month.

See the Wandering Epicycle

**(Update 1/3/2022)** First, this problem is dealt with in more detail and more expansively on the Mathologer Youtube website by Burkard Polster in his 7 December 2018 post on the “Secrets of the Nothing Grinder” (Figure 1). A further, deeper discussion of epicycles is given in the Mathologer’s 6 July 2018 post on “Epicycles, complex Fourier and Homer Simpson’s orbit” (Figure 2). And finally, a panoply of related puzzles is given in the 30 December 2021 Mathologer post “The 3-4-7 miracle. Why is this one not super famous” (Figure 3).

This last post reveals the ambiguity of the idea of “one full (360°) rotation” I disingenuously added to the problem to try to get the answer given in Math Calendar version.

For a complete explanation see the Wandering Epicycle Addendum.

Here is another problem from the 2020 Math Calendar.

As a hint, recall that all the answers are integer days of the month. And the solution employs a technique familiar to these pages.

See Autumn Sum

This is another problem from the 2020 Math Calendar.

“Find the difference between the highest and lowest roots of

*f*(*x*) = *x*^{3} – 54*x*^{2} + 969*x* – 5780”

See Root Difference

Here is another problem from the 2020 Math Calendar to stimulate your mind.

Remember that the answers to Math Calendar problems must all be whole numbers representing days of the month.

See New Years Sum

This is a surprisingly challenging puzzle from the *Mathematics 2020* calendar.

“The sketch is of equally spaced railroad ties drawn in a one point perspective. Two of the ties are perceived to the eye to be 25 feet and 20 feet respectively. What is the perceived length x of the third tie?”

Even though the ties are equally-spaced and of equal length in reality, from the point of view of perspective they are successively closer together and diminishing in length. The trick is to figure out what that compression factor is. I had to review my post on the Perspective Map to get some clues.

See the Railroad Tie Problem