There is Guess Number game like this: In this game, the player must find a hidden positive number by at most $T$ guesses (or turns). The parameter $T$ together with a health parameter $H$ is determined at the beginning of the game. In each turn, the player must say a number. If the number is equal to the hidden number, he wins provided that $H≥0$. If the number is bigger than the hidden number, $H$ is decreased by $1$ unit of health. Otherwise, $H$ remains unchanged. When $H$ becomes negative or $T$ reaches $0$, the player definitely loses. The player can see the remaining turns and units of health after each turn.
How can I find the smallest $M$ for which at least a number from $1$ through $M$ as the hidden number can’t be guessed for the given $T$ and $H$. For example, there is not any ways for finding all positive integers not greater than $M=3$ by $2$ turns and $0$ units of health. Another example: $H = 1$, $T = 3$ So the answer is $M = 7$