Mathematical Puzzles

[Rudolf]

I will try to present mathematical puzzles that I find interestingif you know any or have questions please post

[Rudolf]

Belt around the world

Imagine a belt along the equator of the earth, about 40,000kilometers=40.000,00meters. Every 10meters a person is standing, 4.000,000persons in all. Then the belt is widened 10meters to 40,000.01km=40.000,010meters. Now those 4m persons distribute the widened belt equaly and try to lift the belt away from the surface of the earth. How high can they go?

[Rudolf]

Money distributions

3 persons decide to buy a boat together. The seller tells them the prize is $15,000. They agree and each of them pays him $5,000. They leave, but later his boss finds out that the real prize was only $10,000, so he sends him with the $5,000 after the buyers. As $5,000 are hard to divide by 3 he decides to keep $2,000 for himself an give each of his 3 customers $1,000, so that they have each effectively paid $4,000. Together they now paid $12,000 and the dishonest seller has kept $2,000. Where are the missing $1,000?

(Similiar puzzles like this let some people doubt the value of mathematics in real world problems.)

[The Creator]

Money distributions

3 persons decide to buy a boat together. The seller tells them the prize is $15,000. They agree and each of them pays him $5,000. They leave, but later his boss finds out that the real prize was only $10,000, so he sends him with the $5,000 after the buyers. As $5,000 are hard to divide by 3 he decides to keep $2,000 for himself an give each of his 3 customers $1,000, so that they have each effectively paid $4,000. Together they now paid $12,000 and the dishonest seller has kept $2,000. Where are the missing $1,000?

(Similiar puzzles like this let some people doubt the value of mathematics in real world problems.)

Ah, but I don't think, unless I'm overlooking another way to solve this, that multiplying the $4,000 they each have by 3 ($12,000 total) then adding that $2,000 would give you the right result, because you must consider they each had $5,000 to begin with. That's $15,000 total. The real price was $10,000, so if the seller kept $2,000 and gave each of them $1,000 in return, then they each spent $4,000, but they also each still have $1,000. (($4,000 X 3) + (3 X $1,000) = $15,000). So no missing $1,000.

Or another way to look at it:

All 3 spent $5,000. Then the seller realized that was $5,000 more than needed, but he was greedy and kept $2,000 for himself then gave $3,000 back. So it's like, if two people had $5,000 ($10,000 total between them), one had $3,000 ($13,000 so far), and he had $2,000 ($15,000 total).

Again, no missing $1,000.

[Moviedweeb]

Money distributions

3 persons decide to buy a boat together. The seller tells them the prize is $15,000. They agree and each of them pays him $5,000. They leave, but later his boss finds out that the real prize was only $10,000, so he sends him with the $5,000 after the buyers. As $5,000 are hard to divide by 3 he decides to keep $2,000 for himself an give each of his 3 customers $1,000, so that they have each effectively paid $4,000. Together they now paid $12,000 and the dishonest seller has kept $2,000. Where are the missing $1,000?

(Similiar puzzles like this let some people doubt the value of mathematics in real world problems.)

They paid $4k but still had another $1k in their pocket so it is not missing (just not being spent).

[Rudolf]

ad Money distribution: you can either look at it from the buyer's or from the seller's point of view,the error is the mixed view - correctad Belt around the world:if you are too lazy to do the math, care to give an estimate?

[Rudolf]

the tallest Dwarf is smaller than tiniest Giant

Imagine a sport festival with a group of people standing in a rectangular formation 10 rows and 20 columns, 200 people alltogether. Now first from every row the tiniest is determined and from this 10persons the tallest - this person is the tallest Dwarf. Next from every column the tallest is determined and from those 20 persons the tiniest - this person is the tiniest Giant. Can you prove that the tallest Dwarf is smaller than tiniest Giant ( with the possible exception that this is the same person )?

[XenoZodiac]

Belt around the world

Imagine a belt along the equator of the earth, about 40,000kilometers=40.000,00meters. Every 10meters a person is standing, 4.000,000persons in all. Then the belt is widened 10meters to 40,000.01km=40.000,010meters. Now those 4m persons distribute the widened belt equaly and try to lift the belt away from the surface of the earth. How high can they go?

~1.6m

[Rudolf]

the next Proof I will present without first giving as a puzzle, because I find it so beautiful

take the numbers from 1 to 101 and write them down in any random order n_i , i=1,2,...101 (this is called a permutation), prove that you can find at least one subsequence of length 11 so that all numbers in this sequence are either all increasing or all decreasing.

example: 10,9,8,7,6,5,4,3,2,1,20,19,18,17,16,15,14,13,12,11,30,29,28,27,26,25,24,23,22,21,40,39,38,37,36,35,34,33,32,31,50,49,

48,47,46,45,44,43,42,41,60,59,58,577,56,55,54,53,52,51,70,69,68,67,66,65,64,63,62,61,80,79,78,77,76,75,74,73,72,71,90,89,88,

87,86,85,84,83,82,81,100,99,98,97,96,95,94,93,92,91,101

with subsequece 10,20,30,40,50,60,70,80,90,100,101 length 11

Proof: attribute to each number i (i=1,2,...101) in the random sequence the ordered pair (a_i,b_i), where a_i is the length of the longest increasing subsequence starting with the number in position i n_i and b_i is the length of the longest dereasing subsequence starting with n_i

now all those ordered pairs must be different because if (a_i,b_i)=(a_j,b_j) for i<j then either n_i <n_j or n_i>n_j, in the first case a_i>a_j because I could start a subsequence with n_i,n_j and continue with the longest increasing subsequence starting with n_j, or in the second case b_i>b_j because I could start a subsequence with n_i,n_j and continue with the longest decreasing subsequence starting with n_j

but with a maximum of 10 only 100 different ordered pairs (a_i,b_i) could be made,so at least one has to contain an a_i or a b_i > 10

quot erat demonstrandum

[Rudolf]

~1.6m

did you know the puzzle before?

[XenoZodiac]

the tallest Dwarf is smaller than tiniest Giant

Imagine a sport festival with a group of people standing in a rectangular formation 10 rows and 20 columns, 200 people alltogether. Now first from every row the tiniest is determined and from this 10persons the tallest - this person is the tallest Dwarf. Next from every column the tallest is determined and from those 20 persons the tiniest - this person is the tiniest Giant. Can you prove that the tallest Dwarf is smaller than tiniest Giant ( with the possible exception that this is the same person )?

I am assuming that none of them are the same height.

Let us assume that excepting the event where Tallest Dwarf (TD) and Shortest Giant (SG) are the same person, TD>SG. Meaning anyone of not equal to TD or SD must be either taller than both of them, OR shorter than both of them OR taller than SG but shorter than TD.

Now, in the same row with the TD and same column with the SG, the person MUST BE taller than TD but shorter than SG. However such a person cannot exist since TD>SG.

So, TD>SG is impossible.

So, shortest giant must always be taller than tallest dwarf.

(Phew!! This took a bit of time to know how to solve it)

[XenoZodiac]

did you know the puzzle before?

No, the only puzzle here that I knew before was the money distribution one.

[Rudolf]

~1.6m

mathematically quite easy (simple geometry) but contre intuition which would sugest that 10meters around the globe would not matter

[XenoZodiac]

mathematically quite easy (simple geometry) but contre intuition which would sugest that 10meters around the globe would not matter

1.6m would not matter to the radius of the earth.

[Rudolf]

again not a puzzle but very interesting and understandable for everyone

Arrow's Parodoxon

democratic decisons are not consistent with simple rules of logic

we have 3 persons A, B and C with their different perferences for flavours of icecream

A prefers vanilla over lemon and lemon over strawberry

B prefers lemon over strawberry and strawberry over vanilla

C prefers strawberry over vanilla and vanilla over lemon

a simple rule of logic is transitivity, meaning if you prefer x over y and y over z, you also have to prefer x over z,else you would not be consistent

so we asume A, B and C are all logically consistent so

A prefers vanilla over strawberry

B prefers lemon over vanilla

C prefers strawberry over lemon

now as a community they democraticaly vote

vanilla over lemon because A and C against B

lemon over strawberry because A and B against C

strawberry over vanilla because B and C against A

and those 3 preferences contratict the law of transitivity

therefor a democratic decision process is illogical

[XenoZodiac]

This is one of my favorites.

FORTY + TEN + TEN = SIXTY

Here, there are 10 letters of English alphabet. (E, F, I, N, O, R, S, T, X and Y)Each of them stands for a unique digit from 0-9 such that the above addition is mathematically correct. Which digit does each letter stand for? Edited July 14, 2012 by XenoZodiac

[Rudolf]

This is one of my favorites.

FORTY + TEN + TEN = SIXTY

Here, there are 10 letters of English alphabet. (E, F, I, N, O, R, S, T, X and Y)Each of them stands for a unique digit from 0-9 such that the above addition is mathematically correct. Which digit does each letter stand for?

nice , one would never think, that the data is enough to lead to a unique solution (almost like a sudoku)

29786850850-------31486

I save myself the trouble to show its unique

[Asyulus]

A tall and rusty farmer had 69 tomatoes, all but 68th tomato lost.  How many tomatoes he has left?