Saturday, September 21, 2002
M I N D  G A M E S

How Kandahar was won
Aditya Rishi

Cross connection

Task: To convey news to at least 14 persons in 3 minutes, using one phone call, with one minute "for each call". Let's call: We could just phone ourselves, so 14 persons to share the news with would take 14 calls. If each call takes just 1 minute, we will be on the phone for at least 14 minutes (if everyone answers the call immediately). We could use speaker phones, so that, others in the room, too, can hear the call. Get at least 2 persons to hear each call. That would halve the number of calls from 14 to 7. Ask each person who receives a call to not only put the call through the speakers, but also to do some calling, too. So if two person hear our message, they could each call two others and pass it on in the same way and so on. In the first minute, my first call is heard by A and B. A's call is heard by both C and D; B's call by E and F, C's call by G and H, D's call by I and J, E's call by K and L; and F's call by M and N. Three calls, three minutes, 14 persons reached and you call just once. The secret lies in Fibonacci series of numbers.

TWO persons, Shakuni and Senapati, play a game on a board divided into 3x100 squares. At stake is the kingdom of Gandhara (modern-day Kandahar) that Senapati has annexed from Shakuni. Shakuni gets Dhritarashtra to announce that Shakuni shall play the first move; and Senapati, thinking himself to be royal, agrees to give Shakuni this advantage (big mistake). Shakuni and Senapati move in turn: the first player places tiles of size 1x2 lengthwise along the axis of the board), the second in the perpendicular direction. The loser is the one who cannot make a move. Which of the players can always win (no matter how his opponent plays), and what is the winning strategy?


Shakuni won; he didn't let his opponent score even once. Shakuni was someone like the match-fixers of today; and, in this case, he rigged the toss (to elect to bat first). The secret of his victory lies in the board and his rigged choice to make the first move. We know how the board looks like, so, let us partition the board into 25 parts of 3x4. The first player's strategy is to put tiles into the middle lines of these parts. For his first move, he chooses any part; if the second player puts his tile into the same part, then, the first player chooses any free part for his next move; otherwise, he puts his tile in the same part like the second player did. This guarantees at least 25 moves for the first player, leaving not more than 25 additional moves for the second player. However, the first player is guaranteed at least 25x2 times the other's moves (above and below his tiles) and the second player cannot prevent him from making these moves. Whoever moves first, wins; also, whoever rigs the toss, gets to play first and win.

Reading Shakuni's mind was so difficult that only one man tried to play this game with him. He is Puneet Goyal of Nabha, who sent us this answer: "Shakuni always tries to keep one empty box that the Senapati has filled. Consider that the chess is made of 3*10 squares.

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10

b1 b2 b3 b4 b5 b6 b7 b8 b9 b10

c1 c2 c3 c4 c5 c6 c7 c8 c9 c10

Suppose a1 represents First Square, a2 second square, a3 third (b1…in second row and c1…in third row) and so on. "First, Senapati makes a move. Sena: He fills a1 a2. Shak: He keeps b1 c1 empty and fills b2 c2. Sena: He fills a3 a4. Shak: He keep b3 c3 empty and fills b4 c4. This process continues and it can be extended to 300 squares. Senapati may fill anywhere, or take second turn, but Shakuni will win if he keeps empty column just ahead of every move made by Senapati. My solution is desi." Letting Senapati play first would surely have ruined Shakuni. Write at The Tribune or