Paradox about high-dimensional spheres!

I have been out of puzzle world for sometime. Leaving IIT Kanpur meant leaving company of theoretical souls such as Ramprasad, Deepanjan and Sagarmoy. Interactions with them drove me to write most of the articles I write here.

Anyways, I had an interesting chat with Ramprasad and pretty much copying the conversation.

Imagine four unit circles centred around the 4 points (1,1) (1,-1)
(-1,1) (-1,-1). All the four circles of course can be enclosed within
a square of side two (corners at (plusminus 2, plusminus 2)). Now
there is this tiny gap around the origin right? Put the largest
possible circle there that touches all these other circles. What is
the radius of this circle?  I won’t insult your intelligence; by
simple geometry it is sqrt(2) – 1.
Now repeat this in 3 dimensions. Unit spheres at all points (plusminus
1, plusminus 1, plusminus 1), and draw the largest sphere centred at
the origin to touch all these sphere. What is its radius now? Again,
easy to see it is sqrt(3) -1.
Now what is the answer to the same question in n-dimensions? It is sqrt(n) – 1.

Okay good. Here is the freaky part. The (higher-dimensional)cube that
encloses all these circles is still of side-length two. The sphere you
drew at the origin has radius sqrt(n) – 1, which is way way larger
than 2 for large n. So the sphere you drew there actually *leaks out*
of the box.

I went insane after first hearing about this. It took me quite a while to get my head around this mathematical fact. Yes, indeed this is a fact, no matter how counter-intuitive it sounds. There is not catch or fallacy here. It’s hard to imagine dimensions beyond 3. But probably here is an explanation why the inner sphere *leaks* out of the box.

If it helps, imagine the following picture. Take a box of side-length
4, and put 4 circles in each of the corners that touches two sides of
the square, but make the radius of the circles something like 1/100.
Now, if you draw a circle centred at the origin but touching these
four 1/100 radius circles, you would see that the circle actually goes
out of the square in the middle of the edge. This I guess is similar
to what happens in higher dimensions. The volume of circles in higher
dimensions is really tiny.

Of course, even in higher dimensions the spheres touch the adjacent sphere which is not the case in the explanation above with circles of radius 1/100. But one can see that the cube side-length remains 4, the main diagonal of the cube is sqrt(16n). Hence, the inner sphere, does not go out of the main diagonal but it very well can *leak*  out of the box from one of its face (or surface or facet, whatever you want to call them).

For N=9, it would be the first time when the inner sphere can not be fully contained within the N-dimensional box that we defined earlier.

Let’s talk about volumes The volume of a sphere of radius r in n dimension is. pi^{n/2} r^n /
Gamma(n/2 + 1)   [Gamma(x) is roughly like the factorial of x, but
defined even all reals. Gamma(x) = xGamma(x-1), and Gamma(1/2) =
sqrt(pi)]. Please see more about Gamma Functions here.

Now, the thing is that the volume of an N-dimensional sphere starts decreasing after the dimension grows above 5 because the denominator increases faster as compared to the numerator in the equation of the volume!! And as N-tends to infinity, the volume approaches zero!!!

My mind has shaken up enough for the day!! That’s why I am going on a 5-day leave starting tomorrow (thanks to easter holidays).

I knew there must be a good reason why I hate geometry 🙂

 

Puzzle : A heavy chessboard!

Well,

I have a thing for dramatic titles! The puzzle that I am about to post here is actually very simple. Without further ado, here it is.

Problem :  There is a standard 8×8 chessboard. Each of its 64 square is assigned a weight. These weights are assigned in such a manner that weight of a square is an average of the weight of the square that it is surrounded by. That is : w(i,j) = w(i-1,j-1) + w(i-1,j) + w(i-1,j+1) + w (i,j-1) + w(i,j+1) + w(i+1,j-1) + w(i+1,j) + w(i+1,j+1)/(num of squares surroinding (i,j)). Of course, the intelligent reader can deduce what would be the formula for the squares lying at the boundary. Now, prove that all the squares have equal weight!

Solution : The solution would require lesser space than the problem description :-). Assume that not all the squares have equal weight. Then there has to be a square whose weight is the least. Weight of this square can not be the average of the weight of the squares surrounding it! Please observe that if there are multiple squares with the least weight, the argument still holds as you will be able to find at least one square which has a neighbour with a higher weight. If not, then all of the squares have the least weight!!

Thanks to Deepanjan for this puzzle!

–Saurabh Joshi

Points on a grid

Hi,

This is again one of the question that tejas faced in the CMI entrance exam.

Problem : Given an nxn grid. You choose 2n-1 points randomly ( each with different co-ordinates on the grid ). Prove that there will be three points amongst these 2n-1 point forming a right-angle triangle.

Solution :

It is easy to see that if we prove that it always have to be the case that there will be a point sharing a column with one point and a row with another point we are done. Looks like good old pigeon hole principle (php) should come to the rescue. Unfortunately, I could not come up with a solution using php.

I used mathematical induction to come up with a proof. And I agree with many other people who say that it does not give much insight into the problem.  Anyways, here it goes.

I am going to prove a general claim.

Claim : If you choose m+n-1 points from an mxn grid, there will be three points forming a right angle triangle.

Base case : 2×2 grid ( does not make sense when m or n is 1 ). This is trivial because we have to choose 3 points.

Also, 2xn is also trivial. We have n+1 points. Using php it is easy to see that there will be two points on the same row and two points on the same column. Because there are only two rows, it has to form a triangle. Similar argument goes for mx2 grid.

Assume that the claim is true for any m-1xn-1 grid.

Now,

(1) choose a point p at (x,y) which does not share either a row or column with any other point.  Remove the column and row denoted by x and y. Now we have m-1Xn-1 grid with m+n-2 (= (m-1) + (n-1) – 1  + 1) points to choose from. Follows from induction hypothesis.

(2) choose a point p at (x,y) which shares the column x ( or row y) with only one another point p1. If p also share a row(or column) with some other point p2, we are done. If not, remove column x and row y. m-1xn-1 grid with m+n-3 = (m-1 +n-1 -1) points to choose from. Follows from induction hypothesis.

(3) There are points p1…pk in the same row ( or column ). One can see that there can not be a point p in any of the column of p1…pk. Hence, remove all columns of p1…pk and the row in question. Now, grid is m-1 X n-k with m+n-1-k ( = m-1 + n-k -1 + 1 ) points to choose from. Follows from the induction hypothesis.

I know that there must be extremely simple proof for this. Arrgh, unable to find it :(.

–Saurabh Joshi

Puzzle : A number game

Well,

This is comparatively a simple puzzle with a nice observation. Given to me by tejas, who in turn, faced it in an entrance exam of CMI.

Problem : You are given integers from 1…2n. You are supposed to divide it in two sub sequence A and B such that, A and B both are of the same size n. A is monotonically increasing and B is monotonically decreasing. Prove that,

for any two such sequences .

Solution : A very simple observation and you have the solution. Let,  . So here is what you do. The moment you observe that A[i] > B[i] for some i, you swap the elements. So the element at A[i] is now in B[i] and vice versa. Thus, you can make sure that set A has all and only elements from 1…n and B has all and only elements from n+1….2n. Note, that the difference is taken over the absolute value, hence swapping does not change it. Now we know that difference between summation of elements of B and A is .

Neat, isn’t it?

–Saurabh Joshi

Puzzle : Black and White hats!

Hi,

This very nice puzzle was brought to me by Ramprasad.

Problem : Set of people ( infinite ) are standing on a line of natural numbers. That is they are standing at position 1, 2, 3 …. and so on. White or black hats are put randomly on all of them. What strategy can they decide on so that either all of them guess the color of their hat right, or none of them get it right?

Solution : I hope you guyz remember another puzzle on the similar lines. In fact, you better read that post and I am going to refer to the strategy employed there. Remember that in that earlier puzzle, they have to come up with a strategy so that only finitely many of them guesses it wrong. So we partitioned set of all strings into groups such that strings belonging to one group are only finite hamming distance away from each other. And then using axiom of choice we have a representative strings C1,C2…. in each group.

Now, the hint is, suppose you are one of the people standing. What additional information do you need to know whether you guessed it right or wrong? Think about it.

Well, if someone tells you the hamming distance of the string S ( in which you belong ) and the representative string Ci, then you know whether you guessed it right or wrong. Because, if you see that S’ ( all but you ) differs from the representative string by distance d, then you have guessed your color right. If you see the difference at d-1 position, then you know that you are in one of the position where two strings differ so you must have guessed it wrong.

There you go! That is the hint. All of them decide on to guess that the string S differs only at d locations where d is odd. Now, if S differs at locations d’ where d’ is even. All those people at position which does not differ from representative string Ci, they will see the difference of d’. But as decided, they will try to make the distance odd, so they will choose the opposite color as given by the representative string. Hence, all of them will guess it wrong. For a person who is in a position which differs from Ci, he will see the difference of d’-1 ( which is odd ) so he will go with the color mentioned in the representative string and thus guesses it wrong.

it is easy to see that if the distance of S is indeed odd, all of them will get it right.

Cool trick, isn’t it? Well, the solution I thought it by myself, with the hint given by Ramprasad.

 

Related posts : here, here, here and here

— Saurabh Joshi

Minimum Spanning Tree

Well,

The other day I came across this problem and ramprasad gave a brilliant solution to it.

Problem : Given a connected weighted graph G, find a minimum spanning tree. But wait, before you jump to any conclusion. The weights of the edges are not real numbers, but a quadratic function of time. So given a time interval [t1,t2] find out the minimum spanning tree T. That is T attains the minimum at time tm.  . Now, find out the minimum spanning tree and the time instant tm at which it attains that minimum value.

Solution : Well, the first solution that popped up in my mind was a brute force approach. Consider all spanning trees. As the weights of edges are quadratic functions, the weights of spanning trees would also be quadratic functions. Now differentiate these functions to find the minima. Now find out the absolute minimum of all these. But apparently, the number of spanning tree of a graph could be as high as   thanks to Kirchoff. So obviously, it does not fit into our perception of “efficient algorithm”.

Ramprasad gave a nice solution which is definitely polynomial time. Not sure whether there are better solutions or not. Note that, prim’s algorithm or kruskal’s algorithm takes into account the order of weights of edge and does not really care about what these actual weights are.  So as long as the order of the weights remains unchanged, you can apply one of these algorithm. Given the time interval [t1,t2] it can give rise to infinitely many instances of the weighted graph. As the weight function of edges are quadratic ( parabola ) there are at most many intersection points. That is, parabola corresponding to two edges intersect at atmost 2 points.  And these intersection points are the points in time where weight of one edge goes beyond the weight of another edge.

So, sort these intersection points by time co-ordinate. Note, that between two consecutive intersection points, the order of the weights remain unchanged and hence one of the standard algorithm can be applied. So we need to run Prim’s or Kruskal’s algorithm (for each such interval giving rise to algorithm overall. Clearly, a major improvement over the brute force method.
Ramprasad and I are not sure whether this can be improved or not. Please leave a comment if you have some improvements in mind.

Secret Sharing-3

As promised in the last post,  I am back with the solutions to the problem we left. There are  couple of solutions when collection of sets is given explicitly. If for example, sets S1 to Sn are given with the constraint that all and only members together from any set should be able to unlock the secret. Given n such sets, there are two solution which are similar in some sense.

First solution, can make the unlocking as hard as possible. That is, there is a brute force way to unlock, even if you are not authorized. In this, we share a secret as n tuple, each co-ordinate in the tuple is of length 1024 ( making it possibilities ). That is for each set, you have 1024 bit. Choose a secret S and make n copies of it. Now, for each element of the tuple ( that is for each set Si) you can use the XOR method that we mentioned in the first post of the series.

Second solution, again uses tuples, but instead of using the XOR one can use the polynomials. Choose, different polynomial for each co-ordinate of the tuple such that the polynomial constant is secret S. This is again described in the first post.
However, when sets are not given explicitly, the size of the tuple becomes very large if one goes on to enumerate all possible sets which can unlock. The solution which ramprasad had in mind is using circuit with threshold gates. is a k-threshold gate which becomes one when k or more of the inputs are set to one.

Now, let us see how the circuit for the counter example mentioned in the last post would look like.

The example was as follows. group of 8 persons. 5 or more should be able to unlock the secret. If there are 4, then they would be able to unlock only if there are even of them from the first half ( in turn, there are even of them from each half 🙂 ).

The circuit would look like this

where LTh and RTh denotes threshold gates for the left and the right half. Note that if we want to enumerate the sets there are sets. In comparision, having such a circuit is a very efficient solution 🙂

Secret Sharing – 2

Well,

We have already talked about secret sharing in one of the earlier post. Funny thing happened recently, Ramprasad came up with a question which is a little bit modified. He had a different answer in mind, but I came up with a very simple solution.

Problem : Given a secret S, you have to divide it amongst such that p1 and p2 together should unlock the secret. Alternatively, p1 or p2 along with three of the other four ( p2 to p6 ) should be able to unlock the secret. However, p2 to p6 together should not be able to unlock the secret.

Solution : I came up with a very simple solution. Take any random polynomial of degree 5 such that the constant term is our secret. Evaluate this polynomial at 10 nonzero random points. Now, give 3 points each to p1 and p2 along with their values. Similarly, distribute one point each to the remaining four persons. See that to construct the original polynomial and hence evaluate it at zero, you need at least 6 points.

For this concrete example this solution is perfectly valid. I was tempted to generalize it as follows.

Let us say that we are given collection of sets. In this case, sets are {1,2},{1,3,4,5},{1,3,4,6} ….. and so on. Meaning, you should be able to unlock the secret when all the members of any set are present. I was tempted to generalize the solution I gave earlier, as follows.

Write it in form of constraints such that we will give each person pi weight wi

So now the constraints will be

( we will give some positive number of points of polynomials to each of them )

( when all members of any set is present it should unlock the secret ( threshold is 1 )

( if not all members of the set are present, they should not be able to unlock the secret )

The reason why threshold is taken as 1 is because once we get any feasible solution to this system of inequalities, we can always scale it up, so that all are positive integers and accordingly we will choose threshold T  ( degree T-1 polynomial ). So when collectively when they have T points, they can construct the polynomial and hence, can unlock the secret.

How do we know that the system of inequalities always have the solution? ( if it does, it has infinitely many of them ). Apparently, it is not always possible to have the solution for such a system. Here, any solution defines a hyperplane in n-dimension ( n being total number of people ) such that all the points above it are acceptable and all the points below it are not.

So for collection of sets which demands non-linear surface, this method will not work. Here is a counter-example that ramprasad came up with.

Let us say that there ar 8 persons. You are supposed to divide the secret in the following manner.

1. If there are 5 or more together, they can unlock
2. If there  are exactly 4 of them, then if there are even number of them from the first half ( p1 to p4 ) then they should be able to unlock the secret.
3. Should not be able to unlock in all other cases.

Observe that this gives rise to non-linear partitioning surface.

There are many different generic solution  to this kind of problem. Will talk about it sometime later 🙂

Circles and Tangents

Recently I came across a very nice puzzle related to geometry.

Problem : Take any three non-intersecting circles of distinct radii. Common tangents of any two will intersect at a point. Prove that the three intersection points due to three pairs of tangents are co-linear.

Solution : Ramprasad and I came up with a proof but we believe that there should be much simple and elegant proof. If you can throw any light, please comment. Here is the proof.

Let the center of circles A, B and C be denoted by . Also the intersection points be denoted by and radii are .

Given any two circles, say A and B. Following relation holds due to congruence of the triangles formed.

. Now,

Simplifying, we will get . Thus, we have got how far a point will be given the radii of two circles and the distance between their centers. Let .

Now, Let be the origin and with respect to that the vectors of be denoted as .

So, vector for will be .

Similarly, and

For three points to be collinear, following must hold for some .

Therefore, we must prove that for some

Simplifying,

and

 

You can verify that it is indeed the case that

 

Very ugly math, and with very high probability I must have made a typo in the calculation. Please try it yourself and convince that the property holds. I am looking for a better proof!

Logic and games -2

Well,

In the earlier post I mentioned about using games as a mean to prove/disprove expressibility of FO ( First Order Logic ).

Let us modify the game that we saw last time. Now each round consists of the following : The spoiler chooses one of the structure, chooses a subset of elements of that structure, duplicator has to do the same on the other structure. Now, the spoiler puts a pebble on one of the elements of the subset he/she has chosen and duplicator replies by playing it on the other structure. We have seen in the earlier post that it is impossible to distinguish between two graph A (two disconnected 5-cycle )and B ( one 10-cycle ) using only 3 pebbles. In fact, you will need log 2k pebbles to distinguish between two disconnected k-cycles and one 2k-cycle.

However, in this modified game, only 3 pebble suffices for the spoiler to win for any k. Here is the strategy for the spoiler to expose the difference between two structure.

Look at the figure. Spoiler first chooses one of the k-cycle in A and puts a pebble. Duplicator selects k-points in B ( 2k-cycle ) and puts a pebble. Now, the spoiler chooses the second k-cycle and puts a pebble and duplicator replies. For the third round, the spoiler chooses to play on B. It selects the shortest path between two fixed pebbles as subset, and puts a pebble right in the middle of it. Duplicator now has to play it in A ( two disconnected k-cycles ). Now, according to pigeonhole principle one of the k-cycle will have two pebbles. As shown in figure, (1,2) and (2,3) are in different cycles. Now on, spoiler will always play in graph B. it will choose a pair of pebbles belonging to different cycles in A, selects the pair having shortest path in graph B and puts the pebble right in the middle of it. The spoiler always maintain that it divides the path between pebbles images of which are in different cycles in the other graph. it is easy to see that, after some time, duplicator will have no answer to spoiler’s move ( isomorphism implied by pebbles can not be maintained in the other graph ).

 

This game gives rise to a logic which is strictly powerful than FO. However, Ramprasad mentions that due to a recent result one can always construct pairs of graphs that spoiler can not win for any k-pebble game. Also, this game has some connection to graph isomorphism but probably ramprasad will give details in the comment 😉

 

Similar games are designed to prove/disprove the expressibility of temporal logic, but may be some other time 🙂

 

— Saurabh Joshi