Why this viral math problem teaches an important lesson

Added: 2026-05-12

[00:00:00]Hey, this is Prewalker.

[00:00:03]Here's a meme that's been circulating for years. An orchestra of 120 players takes 40 minutes to play Beethoven's ninth symphony. How long would it take for 60 players to play the symphony? Let P be the number of players and t be the time playing.

[00:00:21]You would actually be surprised that many students approach the problem in a very bizarre way. They literally just look at the numbers that 120 players take 40 minutes and they think, well, if 120 players take 40 minutes, then when you half the number of players, you would have to double the time. So, it would take 80 minutes to play the symphony.

[00:00:43]This sparked internet outrage and someone made the comment, "That's not how this works. That's not how any of this works." Applying some common sense to the question, the amount of time it takes to play a symphony would be the same regardless of the number of players. If 120 players take 40 minutes, it would take the same amount of time for 60 players. So t would be equal to 40 minutes. If you had to write an equation with the variables t and p, you might write something like t is equal to 40 * p ^0 because for p greater than 0, p ^ 0 is equal to 1. So perhaps we have a mathematical answer to the question.

[00:01:26]But as this problem made the rounds, internet salutes discovered something else that was odd about the question.

[00:01:32]The problem states that it takes 40 minutes to play Beethoven's ninth symphony. And when you look up an article on Beethoven's ninth symphony, you will find out it actually takes much longer. It takes a duration of 65 to 70 minutes. So was this question just a trick question all along so students would learn something about Beethoven's ninth symphony? Partially yes. On Twitter, this question was replied by the teacher herself. Claire Longmore replied, "I wrote this. How did you get this? I'm a math teacher in Noddingham, UK. wrote this 10 years ago. Here is the original whole worksheet. So, this is one of the rare times we know the actual original intent of the person who wrote a viral question. The teacher had a worksheet on direct and inverse proportion. Sort these questions into direct and inverse proportion. Beware, there is one trick question. So, question one reads, Bob puts 20 pounds of patrol in his car and can drive 50 miles before filling up again. How far can he drive on 38 pounds of patrol? Let P be the amount of patrol and D be the distance Bob can drive.

[00:02:42]So many of the questions are pretty similar to this. It's either going to be a direct proportion or an inverse proportion.

[00:02:50]As we get to question five, we have the infamous question about Beethoven's ninth symphony. And all the other questions seem pretty routine to me. The question reached many media outlets around the world and it's a great example of a teacher intentionally putting a trick question to guard against students excessive ro learning of just applying formulas. This is a problem that is unique to math class and I think it's worth going over some of the academic examples. If a ship had 26 sheep and 10 goats aboard, how old is the ship's captain? Anyone with common sense would realize there's not enough information to solve this question. But when students in math class approach it, they often just take a look at the numbers and try to manipulate the numbers in some way. Many students would just take 26 and 10, add them together to get 36, and say the captain's age must be 36 years old.

[00:03:50]This was first shown in a study in France in 1979 and it later came to fame in 2018 when the same question was asked in China and media outlets around the world were amazed that they would ask a trick question to students. It was to guard against excessive rote learning.

[00:04:08]Here's another example. There are 125 sheep and five dogs in a flock. How old is the shepherd? Again, the problem is impossible to solve as stated, but the students just tried to look at the numbers in the problem and come up with some sort of reasonable answer. They took the sum of 125 and 5, but said being 130 years is too old. They took the difference 125 minus 5, but being 120 years is also too large of an age.

[00:04:37]But what about 125 divided by 5? Well, that gives you 25. So the shepherd being 25 years old is just about right and many students gave the answer that the shepherd is 25 years old.

[00:04:51]This is from a 1986 paper in Switzerland and again this has been replicated in many countries.

[00:04:57]Here's another question that was in America. An army bus holds 36 soldiers.

[00:05:02]If 1,128 soldiers are being bus to their training site, how many buses are needed? So the students looked at the numbers in the problem. So we have the number of soldiers and we need to put the soldiers into buses. Each bus holds 36 soldiers.

[00:05:18]So about 75% of students thought, well, we should take 1,128 and divide by 36.

[00:05:25]And they did figure out correctly that this evaluates to 31 remainder 12.

[00:05:33]But did they then apply any common sense to this numerical answer? No, about 29% of students got to this calculation and said the number of buses has to be 31 remainder 12. Only 23% of students realize you needed to round up the numerical answer and they gave the correct answer of 32 buses. This was first shown in a 1983 assessment in America and it's been replicated in the decades since.

[00:06:04]Now, here's a much trickier problem.

[00:06:07]This comes from a Russian mathematician who came to America and lamented about the excessive use of just applying formulas in geometry class. The hypotenuse of a right angle triangle is 10 in and the altitude dropped onto it is 6 in. Find the area of the triangle.

[00:06:25]So he joked that this is how American students would approach the problem.

[00:06:29]They would think about a right triangle that has a hypotenuse of 10 and then you would drop an altitude onto the hypotenuse.

[00:06:36]So now how do you figure out the area?

[00:06:38]Well, we have the formula that the area is 12 b * h. The base is equal to 10.

[00:06:44]The height is equal to 6. So applying this formula, we have 12 10 * 6 and that gives the result of 30 in squared.

[00:06:54]So all the American students were able to calculate an answer. However, there is something conceptually wrong in this answer. He joked that Russian students were not able to come up with any answer at all. Why is that? Russian students had a stronger conceptual understanding of geometry beyond just the formulas. If you have a right triangle with a hypotenuse of 10, then its circumcircle will have the hypotenuse as its diameter. So the radius of the circle will be half the length of the hypotenuse which will be equal to half of 10 or five. So the radius of this circle will be equal to five. The altitude dropped upon the hypotenuse of a right triangle cannot exceed the radius of the circle. It has to be less than or equal to five. Giving a dimension of six is an impossibility.

[00:07:49]Now you might find this is a sinister question. But it is always the case whether you're a student or an employee that you should be thinking about the directions that are given to you and whether they actually make any sense.

[00:08:02]Maybe students don't have time to do that in the standardized tests we have.

[00:08:06]But there are systems of education in the world where students do conceptually understand these relationship. Now I will conclude the video with a trick question from none other than Lewis Carol himself of Allison Wonderland fame. For those who don't know, he was also a mathematician. If six cats capture six mice in six minutes, how many are needed to capture 100 mice in 50 minutes? This would seem to be a standard question of proportion. The number of cats times the rate at which they capture mice times the time will be equal to the number of mice captured. So we know that six cats capture six mice in six minutes. And from this equation we can solve the rate of capture is one mouse per six cat minutes.

[00:08:58]So now we just substitute into another equation where we have the result that a 100 mice are captured and we need to have this in 50 minutes. The rate will be the same of one mouse per six cat minutes. And we just need to solve for the variable cats. So we divide both sides of the equation by 50. Multiply both sides of the equation by 6. And this all cancels out in terms of units and we have 100 * 6 / 50. And this works out to be 12 cats. So it would seem like a very easy problem. you would need 12 cats to capture 100 mice in 50 minutes.

[00:09:39]But what happens when you actually think about the question? So six cats can capture six mice in 6 minutes.

[00:09:48]So we take the number of minutes and if we were to double that, we would double the number of mice captured. So six cats would capture 12 mice in 12 minutes.

[00:09:58]Now, if we quadruple the number of minutes, so we go from 12 minutes to 48 minutes, we will quadruple the number of mice caught. So, six cats can capture 48 mice in 48 minutes. Let's now double the number of cats, which will then double the number of mice caught. So, 12 cats can capture 96 mice in 48 minutes.

[00:10:23]So, we now will need these 12 cats to capture an additional four mice to get to 100 mice in the remaining 2 minutes that we have until 50 minutes have elapsed. So, is this going to be something that's possible?

[00:10:39]Well, it all depends on the interpretation of six cats capturing six mice in six minutes. Since cats tend to work individually, it would make sense that we would look at it from the perspective of one cat, a reasonable interpretation is that one cat is capturing one mouse in 6 minutes.

[00:10:59]But now, if it takes 6 minutes to capture one mouse and there's only two minutes remaining, there's not enough time to catch the remaining four mice.

[00:11:08]Even though we have 12 cats, it's still going to take extra time before we can catch those four mice. So we are not going to be able to capture 100 mice in 50 minutes from 12 cats.

[00:11:21]So using this perspective we are going to have to calculate how many cats it will take. If we multiply the number of minutes by 8, we will have one cat is capturing eight mice in 48 minutes. And now let us multiply the number of cats by 13. So the number of mice will also be multiplied by 13. So then 13 cats will capture 104 mice in 48 minutes. And so we will need 13 cats to capture the required 100 mice in 50 minutes. So that would be the answer under this interpretation. But don't worry, Lewis Carol actually went through three other possibilities. What if the cats actually worked as a team? So six cats would capture one mouse in one minute. Or it might be that three cats capture one mouse in two minutes or two cats capture one mouse in three minutes. Well, clearly from cases A and B, there will be enough time for these 12 cats to capture the remaining mice. But when we need two cats working as a pair to capture one mouse in 3 minutes, which is more than the 2 minutes required, we will need another unit of two cats or another pair of cats in order to get to the 100 mice caught in 50 minutes. So in this case, the answer would have to be 14 cats. Carol remarks, "This is a good example of a phenomenon that often occurs in working problems in double proportion. The answer looks all right at first, but when we come to test it, we find that owing to the peculiar circumstances in the case, the solution is either impossible or else indefinite and needing further data." So we can now look on Beethoven's ninth symphony trick question with a renewed perspective.

[00:13:05]When the internet had an outrage, that's not how this works. That's not how any of this works. As if mathematicians don't understand how musical performances work. This is totally false. It was an intentional trick question so that students would be kept on their toes. And that's exactly the type of lesson we need to guard against excessive wte learning.

[00:13:28]Thanks for making us one of the best communities on YouTube. See you next episode of Mind Your Decisions, where we solve the world's problems one video at a Thanks.