Hector's Puzzled

One hundred computers are connected in a 10x10 network grid, as below. At the start exactly nine of them are infected with a virus. The virus spreads like this: if any computer is directly connected to at least 2 infected neighbours, it will also become infected. Will the virus infect all 100 computers? [Solution]

Two players are seated at a square table. The first player places a coin on the table, the second places a coin on the table, and they carry on placing coins one after another, with the only condition being that the coins are not allowed to touch. The winner is the person who places the final coin on the table, meaning that he or she fills the last remaining space between the other coins. The table has to be larger than a single coin, and all the coins placed must be identically sized. One player is always guaranteed to win if they use a certain strategy. Which one is it — the one who starts or the one who goes second? And what is his or her strategy? [Solution]

You’re so tired that you can no longer walk. Luckily, though, you remembered to bring your robot pal along. The robot must be given a list of directions as instructions, and will attempt to read each instruction in order and move that direction. For example, “NNWES” will instruct the robot to move north, north, west, east, south. If an instruction is impossible to carry out (i.e., there is a wall in that direction), the robot will bump into the wall then move on to the next instruction as if nothing happened. The robot can’t do any computation or communicate with you after it leaves — it blindly follows the instructions given until it reaches the finish or it runs out of instructions. However, you can do as much computation as you want before sending the robot on its merry way. (Hint: The robot has lots, and lots, and lots of memory.) What instructions do you feed your robot to guarantee that it will reach the end square somewhere along its journey? Extra credit: What’s an upper bound for the minimum number of instructions you must feed your robot to guarantee that it reaches the end square? [Solution]

You have 12 gold coins — or so you think! One is fake and is known to have a different weight than the others. It could be heavier or lighter; you only know it’s wrong. Using the same simple balance scale, how do you determine the incorrect coin, and whether it is heavier or lighter, in only three weighings? [Solution]

Suppose you have four ropes and a lighter. Each rope burns at a nonconstant rate but takes exactly one hour to burn completely from one end to the other. You can only light the ropes at either of their ends but can decide when to light each end as you see fit. If you’re strategic in how you burn the ropes, how many specific lengths of time can you measure? (For example, if you had one rope, you could measure two lengths of time: one hour, by simply burning the entire rope from one end, and half an hour, by burning the rope from both ends and marking when the flames meet.) Extra credit: What if you had N ropes? [Solution]