# Stimulating Sequence This is another stimulating little problem from the 2022 Math Calendar.

a1 = 1, a2 = 2, …, an+1 = an + 6an-1

x = lim an+1/an   as   n → ∞

Solve for x.”

As before, recall that all the answers are integer days of the month.

# Two and a Half Circles Here is a problem from the 2022 Math Calendar.

“Two small circles of radius 4 are inscribed in a large semicircle as shown.  Find the radius of the large semicircle.”

As before, recall that all the answers are integer days of the month.

As seemed to be implied by the original Math Calendar diagram, I made explicit that the upper circle was tangent to the midpoint of the chord.  Otherwise, the problem is insufficiently constrained.

# Log Jam Here is a tricky little logarithm problem from the 2021 Math Calendar.

“Find x, where

log2(log4(x)) = log4(log2(x))”

As before, recall that all the answers are integer days of the month.

See Log Jam

# Winter Sum 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

# Remainder Problem Here is a challenging problem from the 2021 Math Calendar.

“Find the remainder from dividing the polynomial

x20 + x15 + x10 + x5 + x + 1

by the polynomial

x4 + x3 + x2 + x + 1”

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

See the Remainder Problem

# Wandering Epicycle 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.

# Autumn Sum 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

# Root Difference This is another problem from the 2020 Math Calendar.

“Find the difference between the highest and lowest roots of

f(x) = x3 – 54x2 + 969x – 5780”

# New Years Sum 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. This is a surprisingly challenging puzzle from the Mathematics 2020 calendar.