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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.

The Sprinkler Mystery: Newsletter Challenge (August 2014)

Posted August 01, 2014 12:00 AM
Pathfinder Tags: challenge question

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

An old home's new sprinklers are scheduled to start up at 9 PM but do not. A tech manually turns on the sprinklers from the controller located in the basement boiler room. He hears from upstairs that they are working fine so he closes up the controller, goes upstairs and sees that the sprinklers have stopped working again. What's going on?

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

39 comments; last comment on 08/01/2014
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Space Trail: Newsletter Challenge (July 2014)

Posted June 30, 2014 12:00 AM
Pathfinder Tags: challenge question

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

The path that a spacecraft follows when traveling to the moon is a distorted figure eight, instead of an ellipse surrounding the earth and the moon. Why is this?

And the answer is:

The reason is energy-saving. The figure eight path allows the spacecraft to remain almost equidistant from the centers of the earth and the moon. At this path the gravitational pull of the earth and of the moon attract the ship in opposite directions, so the net force exerted on the craft is minimized, and it is certainly smaller than the force exerted on an elliptical path.

20 comments; last comment on 07/22/2014
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Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

Posted June 01, 2014 12:00 AM
Pathfinder Tags: challenge questions

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

Two cylinders of equal length and radius are set at the same height on a ramp and allowed to roll to the bottom. The first is a solid aluminum cylinder. The second is a hollow lead cylinder with an inner radius slightly more than 3/4 of its outer radius. Assuming the frictional effects are negligible, which cylinder reaches the bottom of the ramp first?

And the answer is:

The solid aluminum cylinder will reach the bottom first.

The key to this problem is that the cylinders are rolling down the ramp. In this case the masses of the cylinders are unimportant, for the same reason that they would be unimportant if the cylinders were just dropped from the same height (neglecting air friction). What matters in this problem is the difference in the moment of inertia of a solid cylinder as compared to a hollow cylinder (cylindrical shell). This difference leads to different accelerations for the respective cylinder's center of mass.

Center of Mass Acceleration of the solid cylinder:

Where g is the acceleration due to gravity (9.8 m/s^2) and θ is the angle of the ramp.

Center of Mass Acceleration of the hollow cylinder:

Where g is the acceleration due to gravity (9.8 m/s^2) and θ is the angle of the ramp.

Since the acceleration of the center of mass of the solid cylinder is greater than the acceleration of the center of mass of the hollow cylinder, the solid cylinder reaches the bottom of the ramp first.

138 comments; last comment on 07/06/2014
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Supercooled Water: Newsletter Challenge (May 2014)

Posted May 01, 2014 12:00 AM
Pathfinder Tags: challenge questions

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

We know that water freezes at 0º C in standard conditions. However, we also know that water can stay in its liquid state well below 0º C in the same conditions; this is called supercooled water. What causes this to happen?

And the answer is:

In order to form ice crystals water needs a nucleating agent. This can be any type of small material (like dust or dissolved air) which causes the deposition of water molecules, creating the crystal. The reason has to do with the minimum energy required to start the creation of the first piece of ice with a minimum given radius so it can grow to larger radius. If the initial radius is smaller than the minimum critical radius needed, ice growth will require lots of energy, making the growth too difficult or impossible. If the initial ice forms on a nucleating agent, growth will be easier because the initial radius may already be larger than the critical radius.

Water without nucleating agents can still form ice if the water molecules meet in certain orientations, but the probability of encountering this molecular orientation increases if the water is below the freezing point. Below the freezing point, the water molecules become less mobile and produce ice crystals more easily.

37 comments; last comment on 06/12/2014
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Birthday Trio: Newsletter Challenge (April 2014)

Posted March 31, 2014 5:01 PM
Pathfinder Tags: challenge questions

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

A school teacher notices that 3 students in his class of 30 students share the same birthday. What is the percent chance of this occurring?

Assume all days of the year are equally likely as a birthday, ignore leap years, and assume the class represents a random sample of birthdays.

And the answer is:

There is a 2.85% chance that three people among 30 share the same birthday.

This problem is the triplet variation of the The Birthday Problem. The equation to solve this problem is:

where W is the number of triplets present in the class (there could be more than one set of triplets), n is the number of students in the class, and m is the number of days in a year. [A. DasGupta, J. Statist. Plann. Inference 130, 377 (2005)]

96 comments; last comment on 04/28/2014
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Is Diet Soda Really Light?: Newsletter Challenge (March 2014)

Posted March 01, 2014 8:00 AM
Pathfinder Tags: challenge questions

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

A can of a diet refreshment (such as a Diet Coke) and a can of regular refreshment (such as a regular Coke) are placed inside a bucket of water. Both cans contain exactly the same volume of liquid and the cans are identical in size and shape. Which can will float higher relative to the other?

And the answer is:

Sweetening regular soft drinks requires large amounts of sugar. The sugar will be dissolved in the liquid with only a small increase in volume. Remember also that the density of sugar water is greater than that of regular water. Diet drinks require artificial sweeteners to sweeten the drink, but a very small amount of the artificial sweetener will suffice. So the diet drink density is slightly heavier than regular water. Therefore the regular Coke will sink, or will at least float lower than the diet drink.

38 comments; last comment on 03/26/2014
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