This page has a variety of mathematics problems/brain teasers which I have come across and solved at different times. They have appeared in several publications dating from antiquity and I have discussed some of them in a column called "Playthemes" I used run for an Indian Science Magazine in the eighties. Brain teasers buffs will know a fair fraction of them, but --- hopefully --- everyone will find something new. I have put them here because I like them. I plan to add more to this list periodically or even make subjectwise pages if/when I get time.

  1. Prove that the area of a cyclic quadrilateral of sides a,b,c,d is A=sqrt[(s-a)(s-b)(s-c)(s-d)]. Of course, things become familiar when d goes to zero.

  2. Take any n points in a plane and let D and d be the greatest and least distances determined by points of this set. Prove that 2D > sqrt(3) [sqrt(n)-1]d.

  3. A right angled tetrahedron ABCD is one in which the three angles at the vertex A are all right angles. Prove that the square of the area of the triangle BCD is the sums of the squares of the areas on the other three sides. [This can be done mentally in 10 seconds with the right approach.]

  4. If a cube is cut into finite number of smaller cubes, prove that at least two of them must be of same size.

  5. In a circumference of circle with center O and radius R, take three points EFG which are the vertices of an equilateral triangle. Draw three circles with centers E,F and G respectively with radius R. These three circles overlap pairwise forming three petals. Find the area of the overlapping region.

  6. Imagine two very long cylinders of unit radius located along the x-axis and y-axis respectively. Find the volume of the overlapping region. [No, it does not require calculus. Moreover the answer does not involve pi.]

  7. Let S be the set of all positive definite integers written in base 10. Remove from it all the integers which have the digit 9 appearing in them. Add up the reciprocals of the remaining ones. Is the sum finite or divergent ? Remember that the sum of the reciprocals of all primes diverges.

  8. (a) A region of area A in a plane is bounded by a simply connected curve of length L. Is there a straightline passing through the region such that it divides both the area and perimeter into two equal parts?

    (b) There are two nonoverlapping regions of arbitrary shapes, in a plane each bounded by simply connected curves. Is there a straight line passing through both and dividing both simultaneously bisecting the areas ? If "no", prove it; if "yes" give the construction.

  9. A decagon ABC...IJ is inscribed in a circle. Show that the length of AD is the sum of the side of the decagon and the radius of the circle. (Sure, it is trivial to do this with trigonometry; but there is an elegant geometrical solution.)

  10. Let ABCD be a convex quadrilateral with perimeter P and the length of the longer diagonal L. What should be the shape of ABCD if (P/L) has the maximum possible value ? [No, it is not a square!]

  11. The diameter of a convex region of perimeter P is defined to be the length of the longest chord in that region. We can define the "pi" for any convex figure by taking the ratio of perimeter to diameter of that convex figure. Prove that this "pi" is bounded from below by 2 (trivial!) and above by actual pi.

  12. Compared to the last one this is surprisingly easy: Show that the area of any convex region of diameter D is bounded from above by (pi/4)D^2.

  13. A circular region of radius 16 cm has 650 points marked inside. You are given a circular ring of inner radius 2 cm and outer radius 3 cm. Show that for any choice of 650 points, you can always place the ring covering at least 10 points.

  14. What is the area of the largest ellipse that can be inscribed in a right angled triangle of sides a,b ? [Yes, there is an nice way of doing it].

  15. Given an acute angled triangle, inscribe in it a triangle such that it has the minimum possible perimeter. [There are elegant geometrical solutions; but if you are a physicist/engineer it should be trivial]

  16. These three are classics but I have come across people who haven't seen them. So ....

    (a) In any triangle prove that the nine-point circle touches the incircle and the three excircles [If you don't know what a nine-point-circle is your education is incomplete].

    (b) In triangle ABC the trisectors of the interior angles A,B,C are drawn making the adjacent ones meet at D,E,F [There is only one sensible way of doing this]. Prove that triangle DEF is equilateral.

    (c) Start with an arbitrary triangle ABC. Draw three equilateral triangles ABX, BCY, CAZ on the outside on each side of the triangle. Show that the centers of these three equilateral triangles form an equilateral triangle.

  17. In triangle ABC, AB=AC, angle A = 20 degrees, a point P is chosen along AC such that AP = BC. Find angle PBC.

  18. Construct a triangle given the three altitudes. [Before you start drawing a triangle with the 3 altitudes as sides, ask yourself: Will the three altitudes of a triangle always form a triangle ?]

  19. (a) A point P is chosen in the plane of a triangle ABC (either inside or outside the triangle) and the perpendiculars are dropped from P to the sides AC and BC. Prove that if x and y are the perpendicular distances then xy is a maximum when P lies on the bisector of angle C.

    (b) Consider next the case when the perpendiculars are dropped from P to all the three sides with lengths x, y and z. Find P such that the product xyz is a maximum. [Sorry! It is not the incenter.]

  20. Construct triangle ABC given the circumcenter, incenter and one excenter.

  21. When a beer can is full of beer its center of gravity is at the geometric center.When the beer can is empty it is again at the geometric center [assume the can is a perfect cylinder and any asymmetry between the top and bottom faces are ignorable]. As one starts to drink the beer, the CG, of course, starts to come down towards the bottom. It must reach a minimum height and rise again. If the beer weighs x grams and the empty can weighs y grams, find the lowest point reached by CG as a fraction of the height of the can.

  22. Two ladders of length a,b are leaning against opposite walls separated by distance x. Their feet rest at the opposite walls. If the height of intersection of the ladders is d, find x in terms of (a, b, d).

  23. Here is a politically incorrect cryptaithm: (EVE/DID)=.TALKTALKTALK..... Each letter stands for a digit and the right hand side is recurring as shown. Find the numbers.

  24. Show that the 13th of a month falls more frequently on Friday than on any other day.

  25. Let us assume you are a distance x units away from certain disaster in your life. Everything you do has a probability p of moving you 1 unit away from disaster and alas, a probability 1-p of taking you towards the disaster. What are your longterm chances of avoiding disaster in your life ? [Comment: Obviously, for p=0 you court disaster while if p=1 you will monotonically move away from it. Do you think p=0.3 persons are doing better than p=0.01 persons in their life ?]

  26. Every young man dating girls has faced this dilemma. "Should one move towards commitment or move away and hope that the next one will be better ?" Assume that: (a) You hope to consider a sample of N girls, one at a time. (b) You can't go back to someone you rejected. So if you pass up (N-1), you are stuck with the Nth for your life who could be a Medusa. At any given time, you have the rankings of all the previous girls you have dated to compare with the current one. What is your optimal strategy to get the best one ? [Ans: For reasonably large N, pass the first [N/e] however good they are and then choose the first one who is better than all those you have passed, without waiting further. If such a candidate doesn't come up, marry the Nth (Medusa) with the satisfaction that you did your best! I was never convinced "e" is natural until I solved this one!]

Quickies

  1. A point P is chosen inside a square ABCD such that angle PAB=angle PBA =15 degrees. Prove that PCD is equilateral.

  2. Rectangle ABCD has AB=3 BC. Points P,Q trisect AB and are ordered A-P-Q-B. Show that angle CAB+ angle CPB =angle CQB

  3. Give 1000 consecutive integers, none of which is a prime number.

  4. Prove that there exists numbers of the form n=p^q where (a) p and q are irrational and (b) n is rational.

  5. A book of 500 pages has 500 typos randomly distributed. What is the probability that page 29 has no typos ?

  6. (a) A point P is chosen inside an equilateral triangle such that its distances from the vertices are 3,4 and 5 centimeters. What is the size of the equilateral triangle? (b) For a non-quickie generalisation try the following: If a,b,c,d denote the three distances from the vertices and the side of the equilateral triangle show that 3(a^4+b^4+c^4+d^4)=(a^2+b^2+c^2+d^2)^2

  7. In a shuffled deck of cards what is the most probable position [from the top, say] for the first black ace ? What is the most probable position for the second black ace ?

  8. What is the rule behind ordering of these letters: z,x,c,v,b,n,m ?

  9. What about the set : o,t,t,f,f,s,s,e, ..... [Find the next few letters]

  10. Here are sets of numbers enclosed by square brackets. Fill the next set of numbers. [1],[1,1],[2,1],[1,2,1,1],[1,1,1,2,2,1],[3,1,2,2,1,1], [.............]

  11. Given any obtuse angled triangle is it possible to dissect it into smaller triangles, all of which are acute ? [The answer is "yes"]

  12. Express 64 using two 4's without the use of any mathematical symbols.

  13. Add a suitable mathematical symbol in the space between two and three in 2 3 so that the resulting expression has a value between two and three. [I know two ways of doing it]

  14. An old chestnut is to express numbers in terms of, say, 4 fours and mathematical symbols, like 97=4(4!)+(4/4); 71=(4!+4.4)/(.4). Give a general formula to express any positive integer in terms of four 4's and symbols.

  15. 1023 players (yes, not 1024) participate in a tournament in which each game produces a decisive winner. Players are eliminated by knock-out with byes being given when odd number of players occur at any given round. How many matches need to be played to find a winner ?

  16. One of the papers by the famous mathematician Littlewood published in a French journal concludes as follows (The sentences in the journal were, of course, in French; what I give below is the English translation).

    "1. I am greatly indebted to Prof. Risez for translating the present paper.

    2. I am greatly indebted to Prof. Risez for translating the previous foot note.

    3. I am greatly indebted to Prof. Risez for translating the previous foot note."

    Littlewood is completely ignorant of French language; so how did he avoid infinite regression?