3 ms·

My nephew brought home a menacing maths problem

39 points by fjmubeen 10y ago

jack9 10y ago
The closest I can get is 9004 3124 3124 1432 1324
ktRolster 10y ago
Teacher gave the kids an impossible problem. Poor kids, I guess.
- ferrari8608 10y ago
  In theory, it should have been a pretty good exercise for the kids. Where the teacher screwed up was not following up with it as the author said. It could have been a great opportunity, but it seems to me the teacher blew that.
- parksy 10y ago
  The author's objection in this case is that this was just another question on a worksheet that was graded right or wrong then the class moved on to other things without further analysis or explanation. Not that the problem was impossible. If the lesson of an "impossible problem" was a deliberate assignment set with an educational intent then there should have been a follow-up or even a primer beforehand so that the "poor kids" are not left thinking that mathematics is just randomly weird and absurd.
StavrosK 10y ago
> He ultimately could not bring himself to accept that a solution does not exist — this is not how his mathematical world operates. I don't understand this. There is a solution, that the problem is unsolvable. It's as much a solution as finding a valid permutation would have been. It just sounds like his nephew hadn't yet learned that proving unsolvability means solving the problem.
- brianwawok 10y ago
  Not sure they teach this very early in school. Or maybe touch on it a few times but not in depth.
- maxander 10y ago
  But that's not how "math," as taught to and perceived by middle schoolers, works; as far as they've likely ever seen, a math problem is a question for which there exists an answer in the form of a sequence of arithmetical operations and a final number or equality. I doubt these students even recognized the concept of a "proof" in this situation- at best, they simply "saw" that it was impossible. But that "seeing" is simply a piece of everyday human reasoning, not the wholly different things that they associate with math classes and happy math teachers, so of course they feel like something is missing. It does sound like the teacher had a good idea- probably it was precisely to get students out of this notion that every math problem has the simple kind of answer they're used to. But it also sounds like the lesson wasn't taught with attention commensurate with its profundity- its easy to forget that these ideas which are so fundamental for us are still alien to most grade-schoolers (or for that matter, many high-school graduates.)
  - StavrosK 10y ago
    I believe you are exactly correct.
justifier 10y ago
i'd agree with the teacher that this is the most 'mathy' way of getting a kid to do a multi problem worksheet i think it would have been kinder to to do 123 and 900 while also offering the possibility of choosing that it is impossible to achieve i also disagree with the op's tactic of 'proving' this this attempt to do less work to show the impossibility of the problem furthers the math education fallacy that numbers are material stead an abstraction his method would falter in any other base in research this reasoning is great at limiting a problem's scope but is useless as evidence in a proof, for instance: primes must end in 1,3,7, or 9, but to say any number ending in those digits is prime is clearly false..also 2,5 :p i like the question the teacher offered but i think it is more appropriate in the math education i have been touting as the necessary future: teach the student to write a program that builds every possible iteration, adds them, then sorts the sums, then search for your desired value learning math and programming together it's long time to free our thoughts from rote arithmetic so we can think about larger implications and further abstractions more readily i find it difficult to accept that the teacher truly just moved on without discussing this problem, that reads more as an excuse to write this post, but if it is true that is a ridiculous failure on the part of the teacher also, offer kids real open questions stead some trick question, when i was in school i used to say to my math teachers 'why should i do these problems? you already know the answers' they treated me like a jerk, now that mindset is how i direct my research
skierscott 10y ago
Whenever I've seen similar problems in real analysis, the wording of the question always leaves no solution as a possibility. "True or false? Can 1234 be rearranged such that...".
shmageggy 10y ago
The proof is the best answer, but the problem is small enough that you could enumerate the answers on a computer almost instantly. If the proof wasn't convincing enough for someone, this would be easier than manually messing with it. In python: https://gist.github.com/ovolve/77e336ab05fda2aa25ebce8a33677679 https://gist.github.com/ovolve/77e336ab05fda2aa25ebce8a33677...
graycat 10y ago
The teacher misstated the problem. The problem should have asked find a solution or show that there can be no solution. Or prove or disprove. Now the student knows that, really, all problems in math are of the form prove or disprove unless the statement is to prove in which case it is misleading to have the claim false. But, in texts, there can be errors, and a student needs to know that. In college, I was reading a book on group theory and could not confirm a statement in the book. Some hours went by, and I couldn't get it. Eventually we found a counterexample and concluded that the book had an error. Actually, it was just an error in typography, a typo. Later, on a Ph.D. qualifying exam, I struggled too long with a problem and got a failing grade. Yup, the problem asked for a proof, but the claim was false. There was a typo. I appealed, got an oral makeup exam in front of several profs, some angry, and ended with a "High Pass". IIRC, Halmos, Finite Dimensional Vector Spaces just states that all the exercises are of the form prove or disprove. He goes on to say, "then discuss such changes in the hypotheses and/or conclusions that will make the true ones false and the false ones true". At one point one course, for some early homework, my submission was that nearly all the exercises were false -- I'd found that the claims failed on the empty set! Given that the course started out with such sloppy work, I dropped it. Net, really, in general in practice in math, all the statements, due to errors, typos, or whatever, have to be regarded as of the form prove or disprove.
DrScump 10y ago
Perhaps there is a solution in a number system other than base 10. (I'm too lazy to pursue such a proof out of simple curiosity.) From the problem statement, you know that 0 and 9 exist, so that eliminates binary through Base 9. But perhaps this hints at a more broad question: can 9000 be arrived at in any numeric base (> 10)?
- tzs 10y ago
  The sum of digits argument generalizes to all bases. Let the base be b. Then the sum of the unit column digits must be b. The sum of the b's column digits must be b-1, as must be the sum of the b^2's column digits. Finally, the sum of the b^3's column digits must be 8. That gives us a total sum of all the digits of 3b+6. Summing rows instead of columns gives us 1+2+3+4 in each row, and there are 4 rows, so that's a total of 40. Thus, we must have 3b+6 = 40. That has no solution in integers.

39 points by fjmubeen 10y ago

jack9 10y ago

The closest I can get is 9004 3124 3124 1432 1324

ktRolster 10y ago

Teacher gave the kids an impossible problem. Poor kids, I guess.

ferrari8608 10y ago
In theory, it should have been a pretty good exercise for the kids. Where the teacher screwed up was not following up with it as the author said. It could have been a great opportunity, but it seems to me the teacher blew that.
parksy 10y ago
The author's objection in this case is that this was just another question on a worksheet that was graded right or wrong then the class moved on to other things without further analysis or explanation. Not that the problem was impossible. If the lesson of an "impossible problem" was a deliberate assignment set with an educational intent then there should have been a follow-up or even a primer beforehand so that the "poor kids" are not left thinking that mathematics is just randomly weird and absurd.

StavrosK 10y ago

> He ultimately could not bring himself to accept that a solution does not exist — this is not how his mathematical world operates. I don't understand this. There is a solution, that the problem is unsolvable. It's as much a solution as finding a valid permutation would have been. It just sounds like his nephew hadn't yet learned that proving unsolvability means solving the problem.

brianwawok 10y ago
Not sure they teach this very early in school. Or maybe touch on it a few times but not in depth.
maxander 10y ago
But that's not how "math," as taught to and perceived by middle schoolers, works; as far as they've likely ever seen, a math problem is a question for which there exists an answer in the form of a sequence of arithmetical operations and a final number or equality. I doubt these students even recognized the concept of a "proof" in this situation- at best, they simply "saw" that it was impossible. But that "seeing" is simply a piece of everyday human reasoning, not the wholly different things that they associate with math classes and happy math teachers, so of course they feel like something is missing. It does sound like the teacher had a good idea- probably it was precisely to get students out of this notion that every math problem has the simple kind of answer they're used to. But it also sounds like the lesson wasn't taught with attention commensurate with its profundity- its easy to forget that these ideas which are so fundamental for us are still alien to most grade-schoolers (or for that matter, many high-school graduates.)
- StavrosK 10y ago
  I believe you are exactly correct.

justifier 10y ago

i'd agree with the teacher that this is the most 'mathy' way of getting a kid to do a multi problem worksheet i think it would have been kinder to to do 123 and 900 while also offering the possibility of choosing that it is impossible to achieve i also disagree with the op's tactic of 'proving' this this attempt to do less work to show the impossibility of the problem furthers the math education fallacy that numbers are material stead an abstraction his method would falter in any other base in research this reasoning is great at limiting a problem's scope but is useless as evidence in a proof, for instance: primes must end in 1,3,7, or 9, but to say any number ending in those digits is prime is clearly false..also 2,5 :p i like the question the teacher offered but i think it is more appropriate in the math education i have been touting as the necessary future: teach the student to write a program that builds every possible iteration, adds them, then sorts the sums, then search for your desired value learning math and programming together it's long time to free our thoughts from rote arithmetic so we can think about larger implications and further abstractions more readily i find it difficult to accept that the teacher truly just moved on without discussing this problem, that reads more as an excuse to write this post, but if it is true that is a ridiculous failure on the part of the teacher also, offer kids real open questions stead some trick question, when i was in school i used to say to my math teachers 'why should i do these problems? you already know the answers' they treated me like a jerk, now that mindset is how i direct my research

skierscott 10y ago

Whenever I've seen similar problems in real analysis, the wording of the question always leaves no solution as a possibility. "True or false? Can 1234 be rearranged such that...".

shmageggy 10y ago

The proof is the best answer, but the problem is small enough that you could enumerate the answers on a computer almost instantly. If the proof wasn't convincing enough for someone, this would be easier than manually messing with it. In python: https://gist.github.com/ovolve/77e336ab05fda2aa25ebce8a33677679 https://gist.github.com/ovolve/77e336ab05fda2aa25ebce8a33677...

graycat 10y ago

The teacher misstated the problem. The problem should have asked find a solution or show that there can be no solution. Or prove or disprove. Now the student knows that, really, all problems in math are of the form prove or disprove unless the statement is to prove in which case it is misleading to have the claim false. But, in texts, there can be errors, and a student needs to know that. In college, I was reading a book on group theory and could not confirm a statement in the book. Some hours went by, and I couldn't get it. Eventually we found a counterexample and concluded that the book had an error. Actually, it was just an error in typography, a typo. Later, on a Ph.D. qualifying exam, I struggled too long with a problem and got a failing grade. Yup, the problem asked for a proof, but the claim was false. There was a typo. I appealed, got an oral makeup exam in front of several profs, some angry, and ended with a "High Pass". IIRC, Halmos, Finite Dimensional Vector Spaces just states that all the exercises are of the form prove or disprove. He goes on to say, "then discuss such changes in the hypotheses and/or conclusions that will make the true ones false and the false ones true". At one point one course, for some early homework, my submission was that nearly all the exercises were false -- I'd found that the claims failed on the empty set! Given that the course started out with such sloppy work, I dropped it. Net, really, in general in practice in math, all the statements, due to errors, typos, or whatever, have to be regarded as of the form prove or disprove.

DrScump 10y ago

Perhaps there is a solution in a number system other than base 10. (I'm too lazy to pursue such a proof out of simple curiosity.) From the problem statement, you know that 0 and 9 exist, so that eliminates binary through Base 9. But perhaps this hints at a more broad question: can 9000 be arrived at in any numeric base (> 10)?

tzs 10y ago
The sum of digits argument generalizes to all bases. Let the base be b. Then the sum of the unit column digits must be b. The sum of the b's column digits must be b-1, as must be the sum of the b^2's column digits. Finally, the sum of the b^3's column digits must be 8. That gives us a total sum of all the digits of 3b+6. Summing rows instead of columns gives us 1+2+3+4 in each row, and there are 4 rows, so that's a total of 40. Thus, we must have 3b+6 = 40. That has no solution in integers.