White must start with 1. e4. On move 5, white must underpromote to a knight for checkmate.
IM Miodrag Perunovic told me recently that he sent this problem to Garry Kasparov for his students a few decades ago, but none of them could figure it out. In fact, Garry ended up asking Mio directly for the solution after he himself couldn't figure it out within 24 hours! Can anyone here figure it out?
I spent several hours myself on the problem and could not figure it out. However, here are two example ideas of how I can visualize possible underpromotions leading to checkmate. Even though neither of them solve the original problem, maybe they'll help inspire someone to actually find the solution.
[FEN ""] 1. e4 e5 2. Ke2 d5 3. Kf3 dxe4+ 4. Kg3 e3 5. Qf3 e2 6. a3 exf1=N#
There are several issues here, but a crucial problem is that black needs to move both the e and d pawns in order to create the mating net resulting in a mate-in-6.
[FEN ""] 1. d3 e5 2. b3 e4 3. Kd2 exd3 4. Kc3 dxe2 5. Kb2 exd1=N#
The issue here is that black is the one mating white on move 5 and nothing has to do with 1. e4.