Fun maths problems

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Offline Musicforawhile

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Fun maths problems
« on: 07/11/2014 22:16:58 »
Hi, so in the book 'The curious case of the dog in the night-time,' I came across the puzzle about the three doors with the prize behind one door. You are supposed to choose two doors then you should change your initial choice after the first door you pick is a dud. Something like that I think...

Do you know any other maths problems or puzzles that a non-mathematician would enjoy and have a sort of 'fun' context?

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Offline Musicforawhile

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Re: Fun maths problems
« Reply #1 on: 10/11/2014 16:41:18 »
For non-mathemticians like me who like puzzles:

The Puzzle:


There are 2 trees in a garden (tree "A" and "B") and on the both trees are some birds.

The birds of tree A say to the birds of tree B that if one of you comes to our tree, then our population will be the double of yours.

Then the birds of tree B tell to the birds of tree A that if one of you comes here, then our population will be equal to that of yours.

Now answer: How many birds in each tree?


« Last Edit: 13/11/2014 12:36:36 by Musicforawhile »

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Offline alancalverd

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Re: Fun maths problems
« Reply #2 on: 10/11/2014 17:44:16 »
Sorry it has to be equations, but life is too short for words!

Quote
The birds of tree A say to the birds of tree B that if one of you comes to our tree, then our population will be the double of yours.


in other "words"         A + 1 = 2 x (B-1) = 2B - 2   call this "equation 1"

Quote
Then the birds of tree B tell to the birds of tree A that if one of you comes here, then our population will be equal to that of yours.

i.e.       B+1 = A-1   call this "equation 2"

If we subtract 1 from each side of the equation 2, then B = A-2  (equation 3)

Now we have a value for B, so put this value back into equation 1

A + 1 = 2 x (A-2) - 2 = 2A -6

Subtract A from both sides

1 = A - 6

So A = 7

And B = A - 2 according to equation 3, so B = 5
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Offline Musicforawhile

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Re: Fun maths problems
« Reply #3 on: 10/11/2014 18:08:03 »
I take it you're a mathematician [;)]. Those equations were easy enough to follow, thank you.

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Offline Musicforawhile

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Re: Fun maths problems
« Reply #4 on: 10/11/2014 19:40:42 »
The Puzzle:


Find a 10-digit number where the first digit is how many zeros in the number, the second digit is how many 1s in the number etc. until the tenth digit which is how many 9s in the number.


« Last Edit: 13/11/2014 12:36:16 by Musicforawhile »

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Offline David Cooper

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Re: Fun maths problems
« Reply #5 on: 11/11/2014 18:08:43 »
From www.mathisisfun.com

Promotion of a website is one thing, but this link looks as if it goes to a terrorist organisation.

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Offline Musicforawhile

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Re: Fun maths problems
« Reply #6 on: 12/11/2014 00:06:14 »
I am not trying to promote it, I just wanted to cite my source. But I suppose it doesn't really matter, it's not going to be plagiarism really. Terrorism?

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Offline Musicforawhile

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Re: Fun maths problems
« Reply #7 on: 12/11/2014 00:10:36 »
The link was bad if that's what you meant by terrorism. I missed out an 's' by accident.


And I'm not plugging them, I have nothing to do with them. But I didn't make up the puzzle. So just referencing the source.
« Last Edit: 13/11/2014 12:37:34 by Musicforawhile »

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Offline David Cooper

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Re: Fun maths problems
« Reply #8 on: 12/11/2014 17:19:47 »
It wasn't a missing "s", but an extra "i", the result being "isis". As for promoting a website by repeating (or attempting to repeat) a link to the same site so soon in a thread like this, you make it look as if you might be promoting it (even if you actually have nothing to do with the people behind it), and that risks making them look bad, so this is something we always need to be careful to avoid doing.

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Offline alancalverd

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Re: Fun maths problems
« Reply #9 on: 12/11/2014 17:50:39 »
Yes, it's a terrorist organisation. "....intimidation....spreading despair....causing physical harm..."

I've now spent 2 days trying to drive, work, and play music whilst figuring out the "10 digit" puzzle in my head (no, I'm not a mathematician!) and I have come close to maiming or frying most of the population of southern England during the day, then spreading despair among musicians and conductors in the evenings. Tomorrow I'm going to sit down with a pencil and paper and do it properly, because I have no intention of flying whilst my mind is so preoccupied. 

The stain of guilt lies with the perpetrator. May the Lord have mercy on your soul, Sir or Madam.
« Last Edit: 12/11/2014 23:19:17 by alancalverd »
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Offline jeffreyH

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Re: Fun maths problems
« Reply #10 on: 13/11/2014 01:23:12 »
Yes, it's a terrorist organisation. "....intimidation....spreading despair....causing physical harm..."

I've now spent 2 days trying to drive, work, and play music whilst figuring out the "10 digit" puzzle in my head (no, I'm not a mathematician!) and I have come close to maiming or frying most of the population of southern England during the day, then spreading despair among musicians and conductors in the evenings. Tomorrow I'm going to sit down with a pencil and paper and do it properly, because I have no intention of flying whilst my mind is so preoccupied. 

The stain of guilt lies with the perpetrator. May the Lord have mercy on your soul, Sir or Madam.

6210001000
Fixation on the Einstein papers is a good definition of OCD.

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Offline alancalverd

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Re: Fun maths problems
« Reply #11 on: 13/11/2014 08:27:27 »
The world is a safer place, music lovers will be grateful, and perhaps I'll earn some money today instead of gazing vacantly into space.

I was getting there by inspiration and elimination (though the bandleader called it "gasping and farting in the tuba section"), but it would be interesting to see an algebraic solution.
« Last Edit: 13/11/2014 08:29:36 by alancalverd »
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Offline CliffordK

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Re: Fun maths problems
« Reply #12 on: 13/11/2014 10:22:09 »
Ok,

Here is one of my favorite math-ish puzzles that I bumped into a while ago.

What is the next letter in the sequence?

FMAMJJA

(once you have the right answer, you will KNOW that it isn't a guess).  Thus, no need to post the answer either (and definitely not the "key").

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Offline Musicforawhile

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Re: Fun maths problems
« Reply #13 on: 13/11/2014 12:41:27 »
The stain of guilt lies with the perpetrator. May the Lord have mercy on your soul, Sir or Madam.

Is this humour..?

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Offline jeffreyH

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Re: Fun maths problems
« Reply #14 on: 13/11/2014 22:33:03 »
So what is the next letter in this sequence?

EEAAUA
Fixation on the Einstein papers is a good definition of OCD.

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Offline alancalverd

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Re: Fun maths problems
« Reply #15 on: 13/11/2014 22:54:50 »
The stain of guilt lies with the perpetrator. May the Lord have mercy on your soul, Sir or Madam.

Is this humour..?

It certainly ain't science, nor great literature, so, my dear Watson, we are inevitably led to the conclusion that it is what earth-people call humour.
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Offline Musicforawhile

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Re: Fun maths problems
« Reply #16 on: 14/11/2014 15:26:27 »
Can you give a clue with the FMAMJJA puzzle?

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Offline David Cooper

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Re: Fun maths problems
« Reply #17 on: 14/11/2014 17:35:35 »
I thought FMAMJJA looked familiar, and now I remember why. Add another three letters and a five-letter name appears.

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Offline jeffreyH

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Re: Fun maths problems
« Reply #18 on: 14/11/2014 20:46:31 »
You need to add another 5 letters.
Fixation on the Einstein papers is a good definition of OCD.

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Offline David Cooper

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Re: Fun maths problems
« Reply #19 on: 14/11/2014 20:50:26 »
Two of the five are already there.

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Offline Musicforawhile

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Re: Fun maths problems
« Reply #20 on: 14/11/2014 21:06:23 »
The answer to the ten-digit problem is:

6210001000

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Offline Musicforawhile

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Re: Fun maths problems
« Reply #21 on: 14/11/2014 21:08:10 »
Is it a number puzzle or letter puzzle?

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Offline alancalverd

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Re: Fun maths problems
« Reply #22 on: 14/11/2014 23:16:33 »
The answer to the ten-digit problem is:

6210001000


True enough, but is there a logical or algebraic approach to the solution?
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Offline David Cooper

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Re: Fun maths problems
« Reply #23 on: 14/11/2014 23:22:27 »
Is it a number puzzle or letter puzzle?

FMAMJJA should not be approached as a mathematical problem.

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Offline chiralSPO

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Re: Fun maths problems
« Reply #24 on: 14/11/2014 23:47:10 »
Is it easier if posed as JFMAMJJA... ?

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Offline alancalverd

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Re: Fun maths problems
« Reply #25 on: 15/11/2014 00:11:44 »
If you haven't already seen it, may I recommend The Imitation Game? I've just got back from the cinema, deeply impressed by a story well told (even though it's been told many times before). 
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Offline CliffordK

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Re: Fun maths problems
« Reply #26 on: 15/11/2014 09:17:48 »
The answer to the ten-digit problem is:

6210001000


True enough, but is there a logical or algebraic approach to the solution?
You can arrive at the solution through logic.

Consider that you have 10 digits.  9 of them zeros.

9,000,000,000   However, in this case you must have one - 9.
9,000,000,001   However, here you only have 8 zeros, and one 8.
8,000,000,010   However, you also have one 1 (which would leave you with 7 zeros).
7,100,000,100   However, in this case, you have two - ones.
7,200,000,100   However, you now have one two, and thus 6 remaining zeros.
6,210,001,000   Which is your answer.

Now, the question would be whether this answer is unique. 
I believe that it is.  Now, proving that it is unique is more difficult, but one seems to get nowhere by adding in a three.

Algebra is difficult because it doesn't deal with whole numbers as smoothly. 

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Offline alancalverd

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Re: Fun maths problems
« Reply #27 on: 15/11/2014 10:42:22 »
That is the trial and error procedure I was doing in my head when I should have been concentrating on driving, though it would have taken me longer as I was working upwards from 1000000000. Pretty much the crunching approach of Colossus!

My interest is in whether a "direct solution" exists, and then if it is (a) unique and (b) generalisable. Any number theorists out there? 
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Offline CliffordK

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Re: Fun maths problems
« Reply #28 on: 15/11/2014 11:30:27 »
Certainly one of the formulas you could apply is that all the digits must add up to 10.
Given that X0 is the first digit, X1 is the second digit, ... X9 is the last (10th) digit, then one gets the two equations (I think).

X0 + X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 = 10.

0*X0 + 1*X1 + 2*X2 + 3*X3 + 4*X4 + 5*X5 + 6*X6 + 7*X7 + 8*X8 + 9*X9 = 10

Given that all of your integers are positive, then you'll be limited very quickly.

Looking at the two formulas with the answer (6,210,001,000), one gets:
6+2+1+(0+0+0)+1+(0+0+0) = 10
0+2+2+(0+0+0)+6+(0+0+0) = 10

It then reduces to 6 + 2 + 2
Introducing a 3 or a 4 would throw it out of kilter.  I'm having troubles thinking of a better way to put it.

Hmmm....  So, if one was doing Octal:
01234567
42101000

And Hex
0123456789ABCDEF
C210000000000C000

Hmmm, Base 7
0123456
3211000

Now Base 6
012345
Hmmm, I think one gets stuck with this one.

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Offline jeffreyH

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Re: Fun maths problems
« Reply #29 on: 15/11/2014 23:52:43 »
For 0 + 1*X1 + 2*X2 + 3*X3 + 4*X4 + 5*X5 + 6*X6 + 7*X7 + 8*X8 + 9*X9 = 10 certain positions are limited to the value of 1. That is 9, 8, 7, and 6 can only appear once. For 5 we still could't have 2 because we would need a 1 in the twos position etc. Summing to 10 cannot take this into account.
Fixation on the Einstein papers is a good definition of OCD.