Let

be a positive integer. A
Japanese triangle consists of

circles arranged in an equilateral triangular shape such that for each
, the

row contains exactly

circles, exactly one of which is coloured red. A
ninja path in a Japanese triangle is a sequence of

circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with
, along with a ninja path in that triangle containing two red circles.
In terms of
, find the greatest

such that in each Japanese triangle there is a ninja path containing at least

red circles.
proposed by
Merlijn Staps, Netherlands