
My nephew brought home a menacing maths problem - fjmubeen
https://medium.com/@fjmubeen/my-nephew-brought-home-this-menacing-maths-problem-e8bbba30e5cb#.gywzjj8kv
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StavrosK
> 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.

~~~
maxander
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.)

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StavrosK
I believe you are exactly correct.

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shmageggy
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/77e336ab05fda2aa25ebce8a33677...](https://gist.github.com/ovolve/77e336ab05fda2aa25ebce8a33677679)

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graycat
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_.

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justifier
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

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DrScump
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
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.

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skierscott
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...".

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

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ferrari8608
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.

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jack9
The closest I can get is 9004

3124

3124

1432

1324

