DECEMBER 2020 PUZZLE - "A π and 5 Surprise"

Submitted by Ben Sparks / Numberphile

Consider a small number x that is 1 over n 5s; for example, if n = 3, the number x is 1/555.

The December Bonus:Show that the sine of x, evaluated in degrees, is approximately sin_d(x) = π / 10²⁺ⁿ.

Extra Bonus: Show that the result becomes more accurate as n is increased: [10²⁺ⁿ sin_d(x) – π ] ≈ π / 10ⁿ

Hall of Fame (December Puzzle Solvers):

Earl Dombroski (NM)

NOVEMBER 2020 PUZZLE - "How Far the Moon"

Submitted by Dave Thomas

As this diagram shows, a person viewing the full moon at midnight sees the moon slightly larger than if it was viewed just after rising (dawn) or just before setting (dusk). This is because the moon over the horizon is about the radius of the earth farther from the viewer than is the overhead full moon at midnight.

The November Bonus:Two images of the full moon (one taken at midnight, one at dusk) have been obtained from a digital camera, with the exact same focal length and zoom settings.

If the diameter of the digital image of the full moon at midnight is 2536 pixels, and that of the moon at dusk is 40 pixels smaller, what is the distance to the moon, as a multiple of the radius of the Earth, R_E? How does this compare to the actual value? (There is some inaccuracy in measuring diameters from the photos). [Based on a true story.]

Hall of Fame (November Puzzle Solvers):

Earl Dombroski (NM)
Keith Gilbert (NM)

OCTOBER 2020 PUZZLE - "Social Voting"

Submitted by Dave Thomas

Consider the following statistics about Harris County, Texas (includes Houston) and New York County, New York (Manhattan):

The October Bonus:If all of the registered voters in each county formed a line at their polling place, and each voter kept a social distance of six feet from adjacent voters, how long would the typical line be, in miles, in Manhattan? And in Harris County? Assume an equal number of registered Manhattan voters resides in each of the areas for the 250 polling places.

Double Bonus: If a Federal judge grants an injunction barring enforcement of Gov. Abbott's order limiting Texas counties to one ballot dropoff, and twelve ballot centers can be used in Harris County, and are equally distributed, how long would the typical line be, then?

Hall of Fame (October Puzzle Solvers):

Earl Dombroski (NM)
Rocky S. Stone (NM)
Mike Arms (NM)
Allen Robnett (NM)

SEPTEMBER 2020 PUZZLE - "Weaning Weekly!"

Submitted by Dave Thomas

The September Bonus:A patient has been instructed to wean themselves off of a daily medication. Here is the schedule the doctor has recommended:

• Week 1 - one pill per day as usual
• Week 2 - one pill every other day
• Week 3 - one pill every 3rd day
• Week 4 - one pill every 4th day
• Week 5 - one pill every 5th day
• Week 6 - one pill every 6th day
• Week 7 - stop pills altogether

The September Bonus:Starting with Sunday, the first day of Week 1, how many pills will the patient need to get to Week 7?

Hall of Fame (September Puzzle Solvers):

Rocky S. Stone (NM)
Mike Arms (NM)
Gene Aronson (NM)
Terry Lauritsen (NM)
Keith Gilbert (NM)

AUGUST 2020 PUZZLE - "Bazinga!"

Submitted by Dave Thomas

The August Bonus:Solve for ??

Hall of Fame (August Puzzle Solvers):
Paul Braterman (UK)
Mike Arms (NM)
Rocky S. Stone (NM)
Keith Gilbert (NM)
Terry Lauritsen (NM)
Gene Aronson (NM)

JULY 2020 PUZZLE - "Multiple Protection"

Submitted by Dave Thomas

The July Bonus:

• If a bandana is 60% effective in preventing transmission of the coronavirus, and a paper face mask is 80% effective, how effective is the combination (mask and bandana together)?
• If a bandana provides 40% probability of allowing transmission of the coronavirus, and a paper face mask provides 20% probability, what is the probability of allowing transmission for the combination (mask and bandana together)?
• What is the mathematical relationship of the two answers (A) and (B)?

Hall of Fame (July Puzzle Solvers):
Mike Arms (NM)
Paul Braterman (UK)
Keith Gilbert (NM)
Rocky S. Stone (NM)
Terry Lauritsen (NM)

JUNE 2020 PUZZLE - "Picture-Perfect Puzzle"

Submitted by Dave Thomas

The June Bonus:What is the Answer ??

Hall of Fame (June Puzzle Solvers):

MAY 2020 PUZZLE - "Corona EMT Blues"

Submitted by Dave Thomas

An emergency medical technician has to transport three individuals from a public testing site to a waiting area at the hospital. One of the three, Steve, has never been exposed to the coronavirus Covid19, and is Susceptible to getting the disease. A second person, Cathy, is Contagious with Covid19. The third person, Rachel, has contracted and Recovered from Covid19, but may still be contagious.
The medic can only transport one patient at a time. They can’t leave the susceptible person and the contagious person together in either location, because if the EMT isn’t there to enforce social distancing, the susceptible person could become infected. Likewise, they can’t leave the susceptible person and the recovered person together in either location, because the recovered patient might still be contagious.

The May Bonus:How can the medic safely transport the three patients to the waiting area of the hospital, while maintaining social distancing?

Hall of Fame (May Puzzle Solvers):

APRIL 2020 PUZZLE - "Train's a-Comin'!"

Suggested by Rocky Stone, from "The 125 Best Brain Teasers of All Time - by Marcel Danesi"

A train leaves New York for Washington every hour on the hour. Similarly, a train leaves Washington for New York every hour on the half-hour. The trip takes five hours each way.

The April Bonus:If you are on the train from New York bound for Washington, how many of the trains coming from Washington going toward New York would you pass?

Hall of Fame (April Puzzle Solvers):

Allen Robnett (NM)
Paul Braterman (UK)
Terry Lauritsen (NM)

MARCH 2020 PUZZLE - "Ready, Aim, Fire!"

Submitted by Dave Thomas

A cannon is being used in an artillery exercise to shoot a target some distance away. The cannon is tilted at the maximum-range angle of 45^o; assume gravitational acceleration is g=10m/sec².

In the first attempt, the muzzle velocity was set at 110 m/sec, but the target was over-shot by 210 meters.

The March Bonus:

• What value should the muzzle velocity be set at to hit the target?
• If muzzle velocity is not changed, what angle(s) should the cannon be tilted at to hit the target?

Hall of Fame (March Puzzle Solvers):

Paul Braterman (UK)
Keith Gilbert (NM)
Gene Aronson (NM)

FEBRUARY 2020 PUZZLE - "Measuring a Mast"

Submitted by Dave Thomas

South of Belen, a radio mast can be found near I-25, a short distance to the east of the freeway. The angular size of the mast, measured from the north-bound lane of I-25, is about 22^o. The same angle, measured from the south-bound lane, is about 18^o. The two lanes are 36 meters apart.

The February Bonus:

• How tall is the mast (H)?
• How far is the mast from the north-bound lane (X)?

Hall of Fame (February Puzzle Solvers):

Rocky S. Stone (NM)
Eiichi Fukushima (NM)
Keith Gilbert (NM)
Allen Robnett (NM)
Harold H. Gaines (KS)
Swami Lad (ND)
Gene Aronson (NM)

JANUARY 2020 PUZZLE - "Wasting Away"

Submitted by Dave Thomas

A certain radioactive element, Unobtanium, loses one percent of its mass in a year.

The January Bonus:What is Unobtanium’s half-life?

Hall of Fame (January Puzzle Solvers):

Paul Braterman (UK)
Keith Gilbert (NM)
Rocky S. Stone (NM)
Gene Aronson (NM)
Harold H. Gaines (KS)

DECEMBER 2019 PUZZLE - "Fantastic Football Picks"

Submitted by Dave Thomas

Alissa and Greg are participating in an office NFL pool. This week, with 14 of 16 games played so far, they are tied for first place, with 12 correct picks each. For the remaining two games in the week, Greg and Alissa have chosen different teams for both games.

The December Bonus:(A) What is the probability that Greg and Alissa will be tied at week's end?
(B) If there were four games remaining, and Alissa and Greg picked different teams for all four, what is the probability that Greg and Alissa will be tied at week's end?
(Extra Credit) If there were n games remaining (n = any positive even number), and Alissa and Greg picked different teams for all n, what is the probability that Greg and Alissa will be tied at week's end? (Assume that overtimes will be used until a clear winner emerges for each football game; no ties allowed.)

Hall of Fame (December Puzzle Solvers):

Keith Gilbert (NM)
Paul Braterman (UK)

NOVEMBER 2019 PUZZLE - "Vive la Resistance"

Submitted by Dave Thomas

The November Bonus:if the resistances R in the circuit above all equal 15 ohms, what is the resistance between contacts A and B?

Hall of Fame (November Puzzle Solvers):

Eiichi Fukushima (NM)
Harold H. Gaines (KS)
Allen Robnett (NM)
Keith Gilbert (NM)
Gene Aronson (NM)

OCTOBER 2019 PUZZLE - "Pretty Prime Powers"

Submitted by Dave Thomas

The October Bonus:The October Bonus is a variation on September’s theme:

Solve for X, in 3^X = X⁷.

Hall of Fame (October Puzzle Solvers):

Gene Aronson (NM) Mike Arms (NM)
Harold H. Gaines (KS)

SEPTEMBER 2019 PUZZLE - "Three Ring Circle"

Submitted by Dave Thomas

The September Bonus:

Solve for X, in 3^X = X³.

Hall of Fame (September Puzzle Solvers):

Gene Aronson (NM)
Paul Braterman (UK)
Mike Arms (NM)
Harold H. Gaines (KS)
Zsolt Nagy (Germany)

AUGUST 2019 PUZZLE - "Pricy Trash Run"

Submitted by Dave Thomas

Suppose you have a rocketload of undesirable trash (perhaps radioactive sludge, or vain politicians’ tweets), and you’ve managed to get it off Earth, and into an Earth-like circular orbit (radius R ~ 1.5x10^11 meters, orbital speed ~ 30,000 meters/second ~ √(GM/R)). Now, you want to dump the trash barge right into the Sun.

The August Bonus:
Of the two methods outlined below, which is the most economical approach? Also, estimate how many joules per kilogram of barge will be required for both methods?

(A) Fire the trash barge’s rockets in the opposite direction of its orbit around the Sun, bringing it to a standstill, from which it would plummet directly into the sun. (This is the inner orbit shown above.)

(B) Briefly fire the barge's rockets in the same direction of motion as the rocket, giving it an elliptical orbit with the perihelion near Earth’s orbital radius, and an aphelion near Jupiter's orbital radius (~7.8x10^11 meters). Once the barge reaches aphelion, then fire the rockets in the opposite direction of its orbit around the Sun, bringing it to a standstill, whereupon it will plunge into the Sun. (This is the outer orbit above.)

Note: assume both Earth and Jupiter are considered "elsewhere" in the solar system for the entire duration, and are shown for scale only. They will not affect the barge in its orbit around the sun. In other words, sun and barge are a 2-body problem.

Hall of Fame (August Puzzle Solvers):

Earl Dombroski (NM)

JULY 2019 PUZZLE - "Takes a Licking, Keeps on Ticking..."

Submitted by Dave Thomas

Suppose you could take an Earthly pendulum clock, of pendulum length L = one meter, to (A) the Moon’s surface, or to (B) Jupiter’s surface. Use the figures below to calculate gravitational acceleration on the surfaces of the Earth, the Moon, and Jupiter.

The July Bonus:
How slow or fast would the clock run on the Moon and on Jupiter, compared to Earth?

Hall of Fame (July Puzzle Solvers):

Allen Robnett (NM)
Earl Dombroski (NM)
Keith Gilbert (NM)
Gene Aronson (NM)
Harold H. Gaines (KS)

JUNE 2019 PUZZLE - "Date with Death"

Submitted by Dave Thomas

Our hero, Dr. Normal, must face seven deadly challenges in order to defeat the arch-villain Tweeter-Man. He only has one chance in seven of surviving each challenge.

The June Bonus:
What is the probability that Dr. Normal will die during the ordeal?

Hall of Fame (June Puzzle Solvers):

Earl Dombroski (NM)
Mike Arms (NM)
Keith Gilbert (NM)
Allen Robnett (NM)
Harold H. Gaines (KS)

MAY 2019 PUZZLE - "Three of a Kind"

Submitted by Dave Thomas

The probability of getting three of a kind in 5-card draw poker (52 cards, 4 suits, 13 ranks per suit) is just over 1 in 50.

The May Bonus:
What is the probability of getting three of a kind in 6-card draw poker? Is it more or less than that for 5-card draw?

Hall of Fame (May Puzzle Solvers):

Earl Dombroski (NM)
Keith Gilbert (NM)
Allen Robnett (NM)
Mike Arms (NM)

APRIL 2019 PUZZLE - "Co-ed Debates"

Submitted by Dave Thomas

In the primary election, nine male and seven female candidates are running for president. Rather than have all 16 candidates try to debate at once, the League of Human Voters has decided to hold four debates, with just four candidates in each event. The debate panel foursomes are to be chosen randomly.

The April Bonus:
How much more likely is a balanced panel (two male/two female) than an all-male panel? And how much more likely is an all-male panel than an all-female panel? (Ratio of probabilities for both.)

Hall of Fame (April Puzzle Solvers):

Earl Dombroski (NM)
Allen Robnett (NM)
Mike Arms (NM)

MARCH 2019 PUZZLE - "Running the Bases"

Submitted by Dave Thomas

The March Bonus:
In what base is the equation 2383 + 2473 + 2601 + 2680 = 11357 valid?
And what is the sum, in Base 10?

Hall of Fame (March Puzzle Solvers):

Earl Dombroski (NM)
Mike Arms (NM)
Allen Robnett (NM)
Keith Gilbert (NM)

FEBRUARY 2019 PUZZLE - "Fishy Encounter"

Submitted by Dave Thomas

A fish whose eyes are one meter below the surface of a river spots a boy. The boy's eyes are one meter above the surface. The horizontal distance between the boy and the fish is two meters. The index of refraction of air is n₁ = 1, and that of water is n₂ = 4/3.

The February Bonus: What are the angles of incidence (I) and refraction (R)?

Hall of Fame (February Puzzle Solvers):

Earl Dombroski (NM)
Gene Aronson (NM)
Allen Robnett (NM)
John Geohegan (NM)
Keith Gilbert (NM)

JANUARY 2019 PUZZLE - "Clock Palindromes"

Submitted by Dave Thomas

Sometimes the time of day, in hours:minutes:seconds, can be a palindrome. Examples of such palindromes include 5:29:25 and 10:33:01.

The January Bonus:

• Of the 86,400 seconds of a day, how many are palindromes?
• What is the shortest interval between two palindromes?
• How many palindromes come five minutes after their predecessors?

Hall of Fame (January Puzzle Solvers):

Earl Dombroski (NM)
Mike Arms (NM)
Rocky S. Stone (NM)
Gene Aronson (NM)

Check out the Old Puzzle Archive, HERE!

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