As usual, watch the video for a solution.

How Tall Is The Dog Puzzle?

Or keep reading.

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Answer To How Tall Is The Dog Puzzle?

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

Method 1: Algebra

Let p be the height of the post, and d be the height of the dog. From the diagram we have:

200 + d = p
p + d = 300

Substituting the first equation for p into the second gives:

200 + d + d = 300
200 + 2d = 300
2d = 100
d = 50 cm

Method 2: Visual Dog

Overlapping the two images of the dog will eliminate the dog variable to give the diagram:

We now have:

2p = 300 + 200
p = 250

Using the right diagram in the original figure we have:

p + d = 300

Since p = 250, we must have d = 50 cm.

Method 3: Visual Pole

We can alternately overlap the images of the poles to eliminate the pole variable:

We thus have:

d + 200 + d = 300
2d + 200 = 300
2d = 100
d = 50

Reference

https://x.com/codek_tv/status/2037414684484321603

An orchestra of 120 players takes 40 minutes to play Beethoven’s 9th Symphony. How long would it take for 60 players to play the symphony? Let P be number of players and T the time playing.

While it appears to be a flawed problem, it actually has an important purpose.

As usual, watch the video for a solution.

Internet Can’t Believe This Math Problem

Or keep reading.

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Internet Can’t Believe This Math Problem

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

References

Symphony
An orchestra of 120 players takes 40 minutes to play Beethoven’s 9th Symphony. How long would it take for 60 players to play the symphony? Let P be number of players and T the time playing.

There will be students that mindlessly use a proportion. They may think that 1/2 as many players will take 2 times as long, or 80 minutes.

Of course the time of a symphony will be fixed regardless of the players, so the answer would be T = 40.

This was intentionally a trick question from a teacher Claire Longmoor, who included it in a problem set of about proportions.

Trick questions are a guard against excessive rote learning and mindlessly using formulas. Here are some examples of how widespread this problem is.

Age of captain
If a ship had 26 sheep and 10 goats onboard, how old is the ship’s captain?

In a study in 1979, French researchers found students would just add the numbers to get 26 + 10 = 36 years old. A similar result happened in China in 2018. Of course the answer is there is not enough information.

Age of shepherd
There are 125 sheep and 5 dogs in a flock. How old is the shepherd?

A 1986 paper in Switzerland found students would manipulate the numbers to find a reasonable answer.

125 + 5 = 130 (too old)
125 – 5 = 125 (too old)
125/5 = 25 (about right)

So they would say the shepherd is 25 years old.

Bus remainder
An army bus holds 36 soldiers. If 1128 soldiers are being bused to their training site, how many buses are needed?

A 1983 NAEP assessment in America found students realized they needed to perform a division:

1128/36 = 31 remainder 12

But only 23% of students rounded up to the correct answer of 32 buses. Some replied 31 buses. And a few shockingly wrote 31, remainder 12 buses!

Area of triangle
The hypotenuse of a right-angled triangle is 10 inches, and the altitude dropped onto it is 6 inches. Find the area of the triangle.

The Russian mathematicians Vladimir Arnold joked that American students could determine an answer from formulas. But Russian students could not. Why not?

Full details in a previous post.

Cat and mouse problem
If 6 cats capture 6 mice in 6 minutes, how many are needed to capture 100 mice in 50 minutes? (Lewis Carroll)

This problem comes from the famed author Lewis Carroll of Alice in Wonderland fame, who was also a mathematician (real name Charles Dodgson).

Using the theory of proportions, we would have:

(cats)(time)(rate) = mice
(6 cats)(6 min)(rate) = (6 mice)
rate = (1 mouse)/(6 cat-min)

(cats)(50 min)(1 mouse)/(6 cat-min) = 100 mice
cats = 100 mice(6 cat-min/1 mouse)/(50 min)
cats = 12 cats

But let’s think about scaling our original information.

6 cats capture 6 mice in 6 minutes.

But if we double the number of cats, and multiply the minutes by 8, we would increase the number of mice by 2×8 = 16, so then 16×6 = 96 mice would be captured.

12 cats capture 96 mice in 48 minutes

Until 50 minutes has elapsed, we only have 2 minutes for the 12 cats to capture 4 more mice. Is it possible?

This entirely depends on how the 6 cats are capturing 6 mice in 6 minutes. Carroll goes through 4 possibilities.

As cats operate individually and not in groups, it would be reasonable to think each is working at a rate of:

1 cat capture 1 mouse in 6 minutes

But in this case, none of the 12 cats could possibly capture a mouse in the remaining 2 minutes! In order to surpass 100 mice in 50 minutes, we would need 13 cats. This can be seen by scaling the cats by 13 and the minutes by 8, so the mice caught are 13×8 = 104.

13 cats capture 104 mice in 48 minutes

There are 3 other reasonable interpretations.

6 cats capture 1 mouse in 1 minutes
3 cats capture 1 mouse in 2 minutes
2 cats capture 1 mouse in 3 minutes

In the first 2 cases, having 2 minutes will be enough for the 12 cats to capture an additional 4 mice.

But in the last case, 3 minutes exceed the 2 minutes left, so we would need 2 more cats in the group. That is, 12 + 2 = 14 cats.

Carroll notes this problem illustrates a common issue in proportion questions. At first you might calculate an answer that appears correct, but upon closer inspection of the details it turns out the problem is either impossible or would require additional information to be solved.

It is great the teacher included Beethoven’s 9th symphony as a trick question, as students can fall into the trap of mindlessly applying formulas, and it is a good reminder to think about the question when finding a solution.

References

Symphony
https://twitter.com/dmataconis/status/917496578285490178https://twitter.com/LongmoorClaire/status/918014499071897600
https://time.com/4979608/beethoven-trick-question/
http://mashable.com/2017/10/12/math-problem-out-of-context/#nT07fua4Xkqz
http://www.nottinghampost.com/news/maths-problem-created-nottingham-teacher-623274
https://www.popsugar.com/moms/Impossible-Trick-Math-Question-44141099
https://medium.com/@intersectarian/beethovens-ninth-97ec40ff24f2
https://www.mathnasium.com/math-centers/denville/news/math-teacher-keep-kids-on-their-toes-with-music-question
Age of captain
https://en.wikipedia.org/wiki/Age_of_the_captain
How old shepherd
https://medium.com/student-voices/how-old-is-the-shepherd-the-problem-that-shook-school-mathematics-ad89b565fff#.xb5jcrgzf
https://www.researchgate.net/publication/331454183_I_added_the_numbers_it’s_math_How_sense-making_in_age_of_the_captain_problems_differs_between_a_mathematics_classroom_and_a_language_classroom
Bus remainder
https://www.inference.org.uk/sanjoy/benezet/rote.html
Triangle
http://imaginary.org/sites/default/files/taskbook_arnold_en_0.pdf
http://math.stackexchange.com/questions/1594740/v-i-arnold-says-russian-students-cant-solve-this-problem-but-american-student
Cat and mouse
https://james.fabpedigree.com/thecats.htm
https://archive.org/details/monthlypacket33unkngoog/page/204/mode/2up?q=lewis

Everyone in the world privately has to press a red button or a blue button. If 50% or more press the blue button, then everyone survives. If not, then only people who press the red button survive. What button would you press? Be honest.

As usual, watch the video for a solution.

Solving The Red Button Blue Button Game Everyone Is Talking About

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Answer To Solving The Red Button Blue Button Game Everyone Is Talking About

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

Let’s analyze the decision using the lens of game theory. In this framework, you accept that other people’s choices are out of your control, and you focus on making your best response to each situation.

Let us consider the possible ways that other people might choose.

Case 1: Sufficiently many people pick blue (regardless of your choice, the total will be at least 50% blue)

Whether you pick red or blue does not matter. You, and everyone in the world, will survive. So both red and blue are best responses.

Case 2: Not enough people pick blue (even with your choice of blue, the total will less than 50% blue)

If you pick red, then you survive and everyone that picked blue does not.

If you pick blue instead, then you will perish with everyone else that picked blue.

Red is your only rational choice.

Case 3: Your choice would be decisive to make 50% blue

If you pick red, then you survive and everyone that picked blue does not. So more than half of the world will die by your choice, as if you were Thanos in Avengers.

If you pick blue, then everyone survives. You get to be the hero and save humanity.

You will survive whether you pick red or blue, so some people have said red is a best response. This is short-sighted. No man is an island, and helping other people survive while not harming yourself is a gain.

So blue is the only rational choice in this situation.

Equilibria

Red seems like a selfish choice, but it might not be so nefarious. Imagine everyone else in the world picked red. Then you picking red is a best response, and no one would want to deviate from their choice of red. So this is an equilibrium. And everyone lives! This situation is akin to a game where stealing (blue button) is punishable by death, so everyone picks not to steal (red button). Or perhaps it is like having everyone drive on one side of the road, and you would also want to join the same decision.

If sufficiently many people pick blue (so that changing 1 red to a blue is still 50% blue), then the game is also at an equilibrium point because everyone survives and any individual that deviates would not change the fact that everyone survives. This situation is akin to a public goods game like if enough people pay taxes or get vaccinated then the entire group can survive and benefit.

Of these two pure strategy equilibria, the majority blue outcome seems more realistic as humans are rarely uniform in their decisions, and blue appeals to a cooperative instinct to save others even if they make a different choice.

In a few polls I have seen, about 55 to 60 percent of people do pick blue. Rarely do viral social media posts have substance, but this thought experiment is a perfect example to analyze with game theory.

Reference

Wait But Why Poll
https://x.com/waitbutwhy/status/2047710215265730755