Answers to last issue's Brain Teasers 1. Here is a nice geometry problem. Show that the angle C equals the sum of angles A and B. Answer: There are many ways of solving this problem. One way is to draw in two more squares, as shown in the figure. The sides of the squares are the same as the diagonals of the original three squares. The additional two squares form the rectangle XYTW. Now, notice triangles ZYB and XYW are similar. To see this clearly look at the next figure, where the triangles are drawn again, first side by side, and then inserted one into the other. The vertical sides of the triangles are equal to the side and diagonal of the original squares of the problem. The slanting sides of the triangles are equal to twice the side and twice the diagonal of the original squares. By the property of correspondence, this means that angles B and D are equal. Hence (A+B) = (A+D). Now the angle (A+D) is the angle made by the diagonal of the first square with its base. Angle C is the angle made by the diagonal of the third square with its base. Since the squares are identical, the angles are also identical (in fact, 45 degrees). Hence C = (A+D). But we have proved that B=D. So C=A+B. 2. Coloured weights: This is a variation on an earlier puzzle. There are six weights. One pair is red, one pair blue, and one pair white. In each pair, there is one weight which is heavier. All the heavier ones are the same weight and all the lighter ones are also the same weight. In appearance they all look the same. Using a weighing balance only twice, can you find out which is the heavier one of each pair? From "Mathematical Circus", by Martin Gardner Answer: The trick here is to maximise the information that you can learn in the first weighing. Take one pair of weights of any colour, say white. Put them on each pan of a balance. At the same time, take two weights of two different colours and put them, one on each pan. Now see if the pans balance. There are two possibilities: the pans balance or one pan is heavier. We shall take them case by case. Case 1: The pans balance. The two whites have to be of different weights. Since the pans balance, it means that the lighter white was put on the same pan as a heavier red//blue, and the heavier white was put with the lighter red/blue. Remove the coloured weights carefully; remember which was paired with which white. Now weigh the white ones alone (this is the second weighing). See which white is the heavier one. If the heavier white had been matched with blue, then that blue is lighter. So the blue that was not weighed is the heavier of the pair. Similarly, if the lighter white one had been matched with red, then that is the heavier of the red pair. So you know which is the heavier and lighter of each of the three pairs. Case 2: The pans do not balance. If you stop and think for a moment, this means that the heavier side MUST have the heavier white. If you are still confused, see the table (where L and H stand for light and heavy, and W/B/R stand for white, blue and red) and read on. Let us assume for clarity that the heavier side has white and red weights and the lighter side has white and blue ones. In the table, Pan 1 has the lighter white and Pan 2 has the heavier white. Now, the red and blue that were added can be chosen in four ways: both light (LL), both heavy (HH), one light and the other heavy (LH and HL). The result of all possible weighings are seen in the table. Pan 1=WL+ Pan 2=WH+ Heavier Pan 1 BL RL Pan 2 2 BH RH Pan 2 3 BL RH Pan 2 4 BH RL Equal Apart from the combination where the pans are equal (which is Case 1 and not valid here), all weighings show Pan 2, the pan having the heavier white, to be heavier. Hence the heavier pan has the heavier white. What about the remaining coloured pairs? We have no information, but we have one more weighing left. Take the red weight that was weighed before and weigh it against the blue that has not yet been weighed. The possibilities are listed in the table. Possibility 4 has been left out since it is not valid. Pan 1= Pan 2= Heavier Pan 1 BH RL Pan 1 2 BL RH Pan 2 3 BH RH Equal Each result is different and so you can tell which colour is heavier/lighter. Hence, the heavier of every colour pair can be found.