Infinite Boxes - Answer
Short answer: No strategy can “slightly” improve your chances. The safest play is to open no boxes, which guarantees a win.
Why? The win condition is never opening an empty box; it is not “walk away with the £1 million.” Because the rules let you choose zero boxes, you achieve a 100 % win‑probability simply by doing nothing.
The statement that “there is a possibility the money is in the first ten boxes” offers no probabilistic edge. You have no way of assigning meaningful likelihoods to any subset of boxes. So unless you’re forced to play, the optimal move remains to do nothing. Any strategy that involves guessing introduces risk; and the jump from any uncertain probability to 100% (via inaction) is not “slight.”
Key idea: winning is defined purely by avoiding failure, not by securing the prize money. The optimal strategy is therefore inaction.