To cheer myself up during the twin problems of (1) underperforming sports team XXX and (2) lockdown in state YYY, neither of which shall be named, I am doing daily number puzzles based on the show Letters & Numbers puzzles (Countdown for those in the UK), where the combination is always two large and four small and the target number is the number of new COVID cases reported in YYY. If I miss the exact target three days in a row then I lose. If I can survive long enough to see XXX win their first game then I also win.
A number puzzle consists of 6 given numbers and a target. We can use elementary operations of addition, subtraction, multiplication and division. Numbers can only be used once, but it is not necessary to use all six numbers to achieve the target. Players can nominate a specific ratio of large versus small numbers (e.g. 3 large and 3 small). Small numbers are between 1 and 10 inclusive and can appear at most twice. Large numbers must be one of 25,50,75,100 and cannot be repeated (hence it is not possible to nominate more than four large numbers).
EXAMPLE: assuming a player chooses two large and four small, the six given numbers could be, 2,9,8,2,100,25 and the target is 179. One possible solution is 2 * (100 + 2) - 25 = 179
In the following questions I am concerned with the chances of hitting the target for a single day, assuming the choice is always two large and four small.
Question 1: If the target number is zero, am I guaranteed to survive for that particular day? (no computers) .
Question 2: What is the smallest unattainable positive target number, if an “adversary” gets to choose the two large and four small numbers? (computers are allowed)
