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Challenge Questions

Stop in and exercise your brain. Talk about this month's Challenge from Specs & Techs or similar puzzles.

So do you have a Challenge Question that could stump the community? Then submit the question with the "correct" answer and we'll post it. If it's really good, we may even roll it up to Specs & Techs. You'll be famous!

Answers to Challenge Questions appear by the last Tuesday of the month.

Baffling Beer Glasses: Newsletter Challenge (April 2017)

Posted March 31, 2017 5:01 PM
Pathfinder Tags: beer challenge question optics

This month's Challenge Question: Specs & Techs from GlobalSpec:

Typical beer mugs have thick walls and a thick bottom. This design serves two purposes. First, it makes the mug heavier, so the drinker assumes the beer is “good.” The second reason is to give the impression that the mug holds more beer than it actually does. Why is this so? Why would the volume appear greater than it is?

And the answer is:

This is an illusion caused by refracted light coming from the beer and passing through the glass and then moving into the air. See, for example, the following figure in which a ray leaves the left edge of the beer. When the ray reaches the edge of the mug it bends when it reaches the air near a viewer’s eyes.

When the viewer’s eyes interact with the ray they mentally extend it back into the glass (mug) and conclude that the original start of the ray is to the left of the actual starting point, as shown in the figure. The mug appears to have a bigger diameter, so the drinker assumes they will enjoy more beer than they actually paid for. Of course, the bartender is happy!

27 comments; last comment on 04/25/2017
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Answer: Prime Partition, March 2017 Challenge Question

Posted March 30, 2017 5:46 PM

Question:

If p(n) is the number of partitions of n, defined as the number of ways to write the integer n as a sum of positive integers where the order of the addends is not significant, what n produces the 42nd largest prime p(n)?

Answer:

The 42nd largest prime p(n) is 3513035269942590955686749126214187667970579050845937, which is produced by n = 2508.

The number of partitions p(n), as stated in the question, is defined as the number of ways to write the integer n as a sum of positive integers where the order of the addends is not significant. So, for example, for n = 4, p(4) = 5 as illustrated below:

4

3 + 1

2 + 2

2 + 1 + 1

1 + 1 + 1 + 1

A prime number is a positive integer that has no positive divisors other than 1 and itself. p(n) is prime for certain values of n, the first few of which are 2, 3, 4, 5, 6, 13, 36, 77, 111… corresponding to p(n) equal to 2, 3, 5, 7, 11, 101, 17977, 10619863,...

You can follow the latest top 20 list of largest p(n) primes at http://primes.utm.edu/top20/page.php?id=54.

8 comments; last comment on 04/03/2017
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Puzzling Prime Partition: Newsletter Challenge (March 2017)

Posted February 28, 2017 5:01 PM

This month's Challenge Question: Specs & Techs from IEEE Engineering360:

If p(n) is the number of partitions of n, defined as the number of ways to write the integer n as a sum of positive integers where the order of the addends is not significant, what n produces the 42nd largest prime p(n)?

The answer to this challenge will be posted later this month, right here on CR4.

27 comments; last comment on 04/06/2017
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Tropical Sun: Newsletter Challenge (February 2017)

Posted January 31, 2017 5:01 PM
Pathfinder Tags: challenge questions tropics

This month's Challenge Question: Specs & Techs from IEEE Engineering360:

The Tropics of Cancer and Capricorn are circles of latitude corresponding to the northernmost and southernmost points where the Sun can still be found directly overhead (on the solstices). Those latitudes are 23.43709°N(S) and are considered the delineation between the temperate zones and subtropical zones. Why are they named “Cancer” and “Capricorn?”

And the answer is:

Cancer and Capricorn were the constellations of the zodiac in which the Sun rose on the days of the summer and winter solstices when the tropics were named 2000 years ago. The solstices are the days when the Sun is directly overhead at the respective Tropics, thus the naming convention. It is interesting to note that due to Precession of the Equinoxes, if the Tropics were named in modern times they would be the Tropic of Taurus and Tropic of Sagittarius. Also worth noting is the term Tropic, which is derived from the Greek “trope,” means “turn back” or “change of direction,” which is what happens to the Sun’s northern or southern latitude advancement on the solstices.

6 comments; last comment on 02/03/2017
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Perplexing Propagation: Newsletter Challenge (January 2017)

Posted December 31, 2016 5:01 PM
Pathfinder Tags: challenge questions radio

This month's Challenge Question: Specs & Techs from IEEE GlobalSpec:

Short-wave and AM transmitters are required to lower their transmission power and directivity of the signal, and, in some instances, are forced to go off the air at sunset. The reason for this FCC requirement: At night, the wave signals from these transmitters (like your local AM radio station) travel a longer distance than during the day. Keeping all the station’s parameters the same, why does the night help to increase the range of these radio waves? Does this also happen with FM transmitters?

And the answer is:

AM signals (including short-wave) are sent to space by the transmitter. How far an AM station’s signal travels depends on the station’s power, its transmission frequency, its antenna, the conductivity of the land around the antenna, and the ionospheric refraction. Some of these waves (the ground signals, as shown in the figure) do not go too far because the earth stops them. The signals that are transmitted to the sky (sky signals, in the figure) reach the ionosphere and are refracted back to earth, which in turn refracts them back to the ionosphere; this process lasts until the signal power is exhausted.

The ionosphere is a layer ranging from 30 miles to over 100 miles from the earth’s surface. The ionosphere is a heavily charged layer of ion molecules, and the number of ions is a function of the amount of sunlight received by the ionosphere. During the day, the production of ions increases, but during the night—in the absence of sunlight—the ionospheric ions decrease, thus, decreasing the width of this layer. A smaller ionospheric width means a longer distance from the surface of the earth, and this raises the refracting levels so the radio signal travels farther around the earth. This is a problem because the AM signal from a particular radio station may interfere with a faraway radio station with the same frequency in violation of allowed ranges for each station. This is the reason that at sunset U.S. AM radio stations must: (a) reduce power to decrease the range, and/or (b) directionalize their signals so they are sent in some directions more than others, or (c) go off the air until sunrise the next day.

For FM signals, on the other hand, because of their higher frequencies, ionospheric refraction is negligible. Having a short wavelength (high frequency), FM signals move through the ionosphere without appreciable interactions, as is depicted in the right hand-side figure.

49 comments; last comment on 03/01/2017
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