A math gem in Matilda the Musical
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January 22nd, 2023 at 12:44:13 AM
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In the Disney/Netflix movie "Matilda the Musical" there is a classroom scene at time index 22:22 in which six year old Matilda is revealed to be a math prodigy. But something is wrong with the scene that might interest math enthusiasts.
In the scene, Matilda's teacher asks for a volunteer to "polish off" the chalk board, meaning to erase some unfinished math problems left behind by a previous class of adults. Matilda misunderstands and takes it upon herself to complete the work on the board, astonishing the teacher.
(Sorry, I can't include a screen shot of the full chalk board due to forum rules until I have 20 posts.)
At first glance, the five equations that Matilda writes on the board look like nonsense, as if they were conceived by a lazy crew member who barely remembered their high school math and just tossed in some equations and lingo that no one was meant to examine closely. However, other parts of the movie, particularly the elaborately choreographed musical number that went viral on TikTok, showed tremendous attention to detail, and no signs of cutting corners. So I thought it was worth taking a closer look at the math.
On closer inspection, applying only a bit of high school math, it is apparent that there are some transcription errors in what Matilda writes. But if you figure out the five equations that she was meant to write, it's clear that anyone so young who wrote that sequence of equations would indeed be a math prodigy.
I'll recap the scene below and the challenge will be to determine what Matilda was intended to write. In other words, determine the correct sequence of five equations. More advanced mathematicians might recognize some greater significance of the sequence (if there is something beyond the obvious), or might even be able to come up with the sequence of equations based on what is on the chalk board, but that is beyond my abilities, and not necessary to correct Matilda's work.
On the board is a lesson culminating in the following quadratic equation:
Question: x^2 - 12x + 44 = 9
Answer:
The solutions to that are x=5 and x=7. If young Matilda were to do nothing more than solve that in her head, it would be impressive.
(Sorry, I can't include a screen shot of Matilda's five equations due to forum rules until I have 20 posts.)
But instead she writes five more equations:
x^2 - 18 + 93 = 16
x^2 - 24 + 168 = 25
x^2 - 30 + 257 = 36
x^2 - 36 + 332 = 49
x^2 - 42 + 501 = 64
Her astonished teacher says "Is that correct?"
Matilda replies "Well, yeah. And then I thought I may as well do the rest. You know, the prime numbers. Where x is the square of AB reoccur. So I realized they must have done it knowing it was part of a sequence."
Your first reaction might be the same as mine. Huh? I don't see anything involving prime numbers, and never heard the term "AB reoccur", and all five equations have semi-random imaginary roots, unrelated at first glance to the original equation. What gives?
But the hint that might make it too easy, is that Matilda (the actress or the crew) made some typos in transcribing the equations onto the chalk board, probably a result of copying them from a note card or having them dictated.
What's the actual sequence of equations that Matilda should have written?
There are a few obvious patterns to notice.
All of the coefficients of X^2 are one. All of the next constants (18, 24, 30, 36, 42) increase by four in each successive equation. And the constants on the right sides of the equation are all perfect squares, the significance of which will become important later. I see no pattern in the second sets of constants (93, 168, 257, 332, 501).
The next thing to recognize is that having two constants on the left side of each equation is strange when they could easily be combined, even if there was some reason to preserve the perfect squares on the right sides. If all five equations were meant to be more "normal" quadratic equations, it seems like maybe the "x"s were left off the coefficients of (-18, -24, -30, -36, and -42). That would eliminate the imaginary roots and leave the following equations:
(1) x^2 - 18x + 93 = 16
(2) x^2 - 24x + 168 = 25
(3) x^2 - 30x + 257 = 36
(4) x^2 - 36x + 332 = 49
(5) x^2 - 42x + 501 = 64
Solving for x, these equations DO have prime roots!
Recall the original equation had roots 5 and 7.
Matilda's equations have these roots:
(1) 7,11
(2) 11,13
(3) 13,17
(4) (36 +/- sqrt(164))/2
(5) 19,23
They really are a sequence of equations with pairs of consecutive prime roots.
The only flaw is equation (4). But note that it would follow the pattern and result in 17 and 19 if the constant "332" was supposed to be "372", another simple transcription error.
The answer to the challenge, the five equations that Matilda was meant to write to show her math skills, were these:
(1) x^2 - 18x + 93 = 16
(2) x^2 - 24x + 168 = 25
(3) x^2 - 30x + 257 = 36
(4) x^2 - 36x + 372 = 49
(5) x^2 - 42x + 501 = 64
It's apparent that this is a sequence of second degree polynomial equations that have roots of two consecutive prime numbers p1 and p2 for which p1 is the nth prime and p2 is the (n+1)th prime in which the polynomial is equal to n squared. But what is the significance? Is it simply a sequence, or does it reveal something about prime numbers? And is anyone familiar with the term "AB reoccurs" and its significance to the sequence? I suspect it to be some theorem, or derivation, or novelty involving number theory, but I'm no Matilda.
...just thrilled that the filmmakers included some actual math, and inadvertently created an easter egg for math enthusiasts to think about.
In the scene, Matilda's teacher asks for a volunteer to "polish off" the chalk board, meaning to erase some unfinished math problems left behind by a previous class of adults. Matilda misunderstands and takes it upon herself to complete the work on the board, astonishing the teacher.
(Sorry, I can't include a screen shot of the full chalk board due to forum rules until I have 20 posts.)
At first glance, the five equations that Matilda writes on the board look like nonsense, as if they were conceived by a lazy crew member who barely remembered their high school math and just tossed in some equations and lingo that no one was meant to examine closely. However, other parts of the movie, particularly the elaborately choreographed musical number that went viral on TikTok, showed tremendous attention to detail, and no signs of cutting corners. So I thought it was worth taking a closer look at the math.
On closer inspection, applying only a bit of high school math, it is apparent that there are some transcription errors in what Matilda writes. But if you figure out the five equations that she was meant to write, it's clear that anyone so young who wrote that sequence of equations would indeed be a math prodigy.
I'll recap the scene below and the challenge will be to determine what Matilda was intended to write. In other words, determine the correct sequence of five equations. More advanced mathematicians might recognize some greater significance of the sequence (if there is something beyond the obvious), or might even be able to come up with the sequence of equations based on what is on the chalk board, but that is beyond my abilities, and not necessary to correct Matilda's work.
On the board is a lesson culminating in the following quadratic equation:
Question: x^2 - 12x + 44 = 9
Answer:
The solutions to that are x=5 and x=7. If young Matilda were to do nothing more than solve that in her head, it would be impressive.
(Sorry, I can't include a screen shot of Matilda's five equations due to forum rules until I have 20 posts.)
But instead she writes five more equations:
x^2 - 18 + 93 = 16
x^2 - 24 + 168 = 25
x^2 - 30 + 257 = 36
x^2 - 36 + 332 = 49
x^2 - 42 + 501 = 64
Her astonished teacher says "Is that correct?"
Matilda replies "Well, yeah. And then I thought I may as well do the rest. You know, the prime numbers. Where x is the square of AB reoccur. So I realized they must have done it knowing it was part of a sequence."
Your first reaction might be the same as mine. Huh? I don't see anything involving prime numbers, and never heard the term "AB reoccur", and all five equations have semi-random imaginary roots, unrelated at first glance to the original equation. What gives?
But the hint that might make it too easy, is that Matilda (the actress or the crew) made some typos in transcribing the equations onto the chalk board, probably a result of copying them from a note card or having them dictated.
What's the actual sequence of equations that Matilda should have written?
There are a few obvious patterns to notice.
All of the coefficients of X^2 are one. All of the next constants (18, 24, 30, 36, 42) increase by four in each successive equation. And the constants on the right sides of the equation are all perfect squares, the significance of which will become important later. I see no pattern in the second sets of constants (93, 168, 257, 332, 501).
The next thing to recognize is that having two constants on the left side of each equation is strange when they could easily be combined, even if there was some reason to preserve the perfect squares on the right sides. If all five equations were meant to be more "normal" quadratic equations, it seems like maybe the "x"s were left off the coefficients of (-18, -24, -30, -36, and -42). That would eliminate the imaginary roots and leave the following equations:
(1) x^2 - 18x + 93 = 16
(2) x^2 - 24x + 168 = 25
(3) x^2 - 30x + 257 = 36
(4) x^2 - 36x + 332 = 49
(5) x^2 - 42x + 501 = 64
Solving for x, these equations DO have prime roots!
Recall the original equation had roots 5 and 7.
Matilda's equations have these roots:
(1) 7,11
(2) 11,13
(3) 13,17
(4) (36 +/- sqrt(164))/2
(5) 19,23
They really are a sequence of equations with pairs of consecutive prime roots.
The only flaw is equation (4). But note that it would follow the pattern and result in 17 and 19 if the constant "332" was supposed to be "372", another simple transcription error.
The answer to the challenge, the five equations that Matilda was meant to write to show her math skills, were these:
(1) x^2 - 18x + 93 = 16
(2) x^2 - 24x + 168 = 25
(3) x^2 - 30x + 257 = 36
(4) x^2 - 36x + 372 = 49
(5) x^2 - 42x + 501 = 64
It's apparent that this is a sequence of second degree polynomial equations that have roots of two consecutive prime numbers p1 and p2 for which p1 is the nth prime and p2 is the (n+1)th prime in which the polynomial is equal to n squared. But what is the significance? Is it simply a sequence, or does it reveal something about prime numbers? And is anyone familiar with the term "AB reoccurs" and its significance to the sequence? I suspect it to be some theorem, or derivation, or novelty involving number theory, but I'm no Matilda.
...just thrilled that the filmmakers included some actual math, and inadvertently created an easter egg for math enthusiasts to think about.
June 13th, 2023 at 7:26:11 PM
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I know you posted this months ago, but I only just got around to watching 'Matilda the Musical". As a math lover, I really appreciate you working it out in this thread. Here are some additional things I deduced. Hope it makes sense.
The sequence actually begins with previous questions on the board.
I do find it funny that instead of writing the answer to the last question, Matilda chooses to write more questions. "Polish off" sounds more like finish the work than "create an endless sequence"
The sequence actually begins with previous questions on the board.
Quote:Question 2: x^2 - 5x + 7 = 1
Answer: x=2 x=3
Question 3: x^2 - 8x + 19 = 4
Answer: x=3 x=5
Question 4: x^2 - 12x + 44 = 9
Answer: x=5 x=7
The pattern is as you described where the question is set equal to a square number and the solutions (roots) are consecutive prime numbers.
Starting with the answer, we can work backwards to figure out what the equation for the next question should be. I will call the prime numbers a and b (This is generally the letters mathematicians use for roots/factoring). Since we know the roots will be x=a and x=b, then
(x-a)(x-b) = 0 We can multiply that out
x^2 - (a+b)x + ab = 0 Then add the square number s to both sides
x^2 - (a+b)x + (ab+s) = s
So the pattern for the last/constant term is ab + s. I believe this is the "AB" Matilda is referencing. "Reoccur" I assume refers to the fact that second prime number b reoccurs in the next equation as a. Like how 5 is a solution for question 3 and then reoccurs as a solution for question 4. I will say the exact wording from the scene "x is the square of AB reoccur" doesn't really make sense, but I assume the explanation was simplified for the movie.
Now that we know the pattern, we can formulate the next equations in the sequence like Matilda (with the corrections noted by OP). Add the prime numbers to get the middle term. Multiply the prime numbers to get the last term and then add the square number. (I wrote out each step for the first equation but simplified the rest).
(1) Answer: x=7 x=11
(x-7)(x-11)=0
x^2 - (7+11)x + (7*11) = 0
x^2 - 18x + 77 = 0
x^2 - 18x + 77 + 16 = 0 +16
Question 5: x^2 - 18x + 93 = 16
(2) Answer: x=11 x=13
x^2 - (11+13)x + (11*13) + 25 = 25
Question 6: x^2 - 24x + 168 = 25
(3) Answer: x=13 x=17
x^2 - (13+17)x + (13*17) + 36 = 36
Question 7: x^2 - 30x + 257 = 36
(4) Answer: x=17 x=19
x^2 - (17+19)x + (17*19) + 49 = 49
Question 8: x^2 - 36x + 372 = 49
(5) Answer: x=19 x=23
x^2 - (19+23)x + (19*23) + 64 = 64
Question 9: x^2 - 42x + 501 = 64
Starting with the answer, we can work backwards to figure out what the equation for the next question should be. I will call the prime numbers a and b (This is generally the letters mathematicians use for roots/factoring). Since we know the roots will be x=a and x=b, then
(x-a)(x-b) = 0 We can multiply that out
x^2 - (a+b)x + ab = 0 Then add the square number s to both sides
x^2 - (a+b)x + (ab+s) = s
So the pattern for the last/constant term is ab + s. I believe this is the "AB" Matilda is referencing. "Reoccur" I assume refers to the fact that second prime number b reoccurs in the next equation as a. Like how 5 is a solution for question 3 and then reoccurs as a solution for question 4. I will say the exact wording from the scene "x is the square of AB reoccur" doesn't really make sense, but I assume the explanation was simplified for the movie.
Now that we know the pattern, we can formulate the next equations in the sequence like Matilda (with the corrections noted by OP). Add the prime numbers to get the middle term. Multiply the prime numbers to get the last term and then add the square number. (I wrote out each step for the first equation but simplified the rest).
(1) Answer: x=7 x=11
(x-7)(x-11)=0
x^2 - (7+11)x + (7*11) = 0
x^2 - 18x + 77 = 0
x^2 - 18x + 77 + 16 = 0 +16
Question 5: x^2 - 18x + 93 = 16
(2) Answer: x=11 x=13
x^2 - (11+13)x + (11*13) + 25 = 25
Question 6: x^2 - 24x + 168 = 25
(3) Answer: x=13 x=17
x^2 - (13+17)x + (13*17) + 36 = 36
Question 7: x^2 - 30x + 257 = 36
(4) Answer: x=17 x=19
x^2 - (17+19)x + (17*19) + 49 = 49
Question 8: x^2 - 36x + 372 = 49
(5) Answer: x=19 x=23
x^2 - (19+23)x + (19*23) + 64 = 64
Question 9: x^2 - 42x + 501 = 64
I do find it funny that instead of writing the answer to the last question, Matilda chooses to write more questions. "Polish off" sounds more like finish the work than "create an endless sequence"
