快猫短视频

#42 Persistence

Set by Peter Rowlett

A number鈥檚 persistence is given by the steps needed to reach a single-digit whole number by repeatedly multiplying all its digits together. For 39, for example, we start with 3脳9 to give 27, then 2脳7 is 14 and 1脳4 is 4. This took three steps, so the persistence of 39 is 3. We ended up with 4, so we say the root of 39 using this rule is 4.

What are the persistence and root of 77?

What is the smallest number with a persistence of 1?

What is the largest persistence of a two-digit number?

Solution next week

#41 There are three prisoners

Solution

If the prisoners pick one of them to speak, and to guess randomly, they have a 50 per cent chance of success. To win 75 per cent of the time, they agree that any of them who can see two blobs the same colour will guess the opposite colour, but those seeing two differently coloured blobs won鈥檛 guess. This will guarantee success except when all three blobs are the same colour (25 per cent probability). After overhearing this strategy, the guards will use the same colour of blob. To beat this, the prisoners agree they will treat the blob of a prisoner who flips heads as the actual colour, but the blob of one who flips tails as the opposite colour, thus randomising the blobs in a way the guards can鈥檛 control. The prisoners then follow the usual rules using the modified colours. If any prisoner guessing has flipped tails, they need to reverse the final answer too.