Tintin reaches Rastapopoulos’s island, but they are informed he has moved out, and the situation is pretty explosive there. Alonso, Bianca, and Coco are running for the post of the island head. On the ballot, each voter lists the three candidates in order of their preference. Counting only the first preferences results, dramatically, in a three-way tie. To break the deadlock, the second preferences are counted, but again there is a three-way tie. Alonso notes that, since the number of voters is odd, they can make two-way decisions without ties. He proposes that the voters first choose between Bianca and Coco, then the winner faces Alonso for the position. Bianca thinks it’s a good resolution, since they only want to identify the winner, not the runner-up. Coco disagrees and complains that this is giving Alonso an advantage. Who is right? Assuming the voters never change their preferences, what is Alonso’s chance of winning under his proposed voting system?A. AlonsoB. CocoC. Both are wrongD. None of these

Tintin reaches Rastapopoulos’s island, but they are informed he has

1.	Tintin reaches Rastapopoulos’s island, but they are informed he has moved out, and the situation is pretty explosive there. Alonso, Bianca, and Coco are running for the post of the island head. On the ballot, each voter lists the three candidates in order of their preference. Counting only the first preferences results, dramatically, in a three-way tie. To break the deadlock, the second preferences are counted, but again there is a three-way tie. Alonso notes that, since the number of voters is odd, they can make two-way decisions without ties. He proposes that the voters first choose between Bianca and Coco, then the winner faces Alonso for the position. Bianca thinks it’s a good resolution, since they only want to identify the winner, not the runner-up. Coco disagrees and complains that this is giving Alonso an advantage. Who is right? Assuming the voters never change their preferences, what is Alonso’s chance of winning under his proposed voting system?A. AlonsoB. CocoC. Both are wrongD. None of these
Answer» From the conditions of the problem, the number of voters is 3n for some odd positive integer n. Each of the candidates A, B and C received n first preferences and n second preferences. Let the number of voters who voted A gt B gt C be x and the number of voters who voted `A gt C gt B` be y. Then x + y = n and the size of the remaining voting groups can be easily computed: `A gt B gt C : x, A gt C gt B : y, B gt C gt A: x, B gt A gt C : y, C gt A gt B : x, C gt B gt A: y`. When only comparing B against C, we have `B gt C : x + x + y = n + x, C gt B : y + x + y = n + y`. Similarly (noting the cyclic symmetry), we have `A gt B : n + x, B gt A: n + y, A gt C : n + y, C gt A: n + x`. Since n is odd, `x 6= y. If x gt y`, then B wins against C but goes on to lose against A. If `y gt x`, then C beats B then also goes on to lose against A. Therefore Christie is right, Adrian will always win under the proposed system.

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You and your friend save as much money as possible by participating in the contest and fly to Athens. Exploring Athens you and all other teams come to Parthenon, the temple of Athena, who is the goddess of wisdom and intelligence. You also see many images of an owl. You learn from the guide that it is the “Athena Noctua”, also called the Minerva owl or the little owl. The guide also mentions some interesting stories of Athena’s wisdom and cleverness. The tourist guide at Parthenon takes you to an abandoned tunnel. And here you find an inscribed stone and a bowl of colored flowers. Suddenly you hear a deep and ancient gong ringing as you start reading the Stone Of Instructions. The instructions read as follows: Every person entering the Temple of Athena will be given a bowl with 15 red flowers and 12 yellow flowers. Each time the Gong of Time rings, he/she must do one of two things: 1. Exchange: If he has at least 3 red flowers in his bowl, then he may exchange 3 red flowers for 2 yellow flowers. 2. Swap: He may replace each yellow flower in his bowl with a red flower and replace each red flower in his bowl with a yellow flower. That is, if he starts with i red flowers and j yellow flowers, then after he performs this operation, he will have j red flowers and i yellow flowers. [You can leave the tunnel only when you make all the combinations (no. of red, no. of yellow) of flowers that are possible]. What is the number of combinations you need to make to leave the tunnel?A. 27B. 52C. 54D. None of these
You and your friend save as much money as possible by participating in the contest and fly to Athens. Exploring Athens you and all other teams come to Parthenon, the temple of Athena, who is the goddess of wisdom and intelligence. You also see many images of an owl. You learn from the guide that it is the “Athena Noctua”, also called the Minerva owl or the little owl. The guide also mentions some interesting stories of Athena’s wisdom and cleverness. The tourist guide at Parthenon takes you to an abandoned tunnel. And here you find an inscribed stone and a bowl of colored flowers. Suddenly you hear a deep and ancient gong ringing as you start reading the Stone Of Instructions. The instructions read as follows: Every person entering the Temple of Athena will be given a bowl with 15 red flowers and 12 yellow flowers. Each time the Gong of Time rings, he/she must do one of two things: 1. Exchange: If he has at least 3 red flowers in his bowl, then he may exchange 3 red flowers for 2 yellow flowers. 2. Swap: He may replace each yellow flower in his bowl with a red flower and replace each red flower in his bowl with a yellow flower. That is, if he starts with i red flowers and j yellow flowers, then after he performs this operation, he will have j red flowers and i yellow flowers. [You can leave the tunnel only when you make all the combinations (no. of red, no. of yellow) of flowers that are possible]. After how many operations will you have 5 each of red and yellow flowers?A. 5B. 10C. 15D. None of these
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