real number to integer points game

Question

Not sure what this question is about, and looking for ideas about a solution. Thanks!

Answer 1

Hint: let me take Jack's place and you take Colin's place. If I play $0.95$, what do you play? How about $9.95?$ What does that tell you? If I play $0.51, 0.49, 0.26?$ Does this help?

Added: We only care about the fractional part of Jack's number, so we will assume the number is in $[0,1)$. We will also assume that you round toward an integer if possible, so $0.05$ rounds to $0$ and $0.95$ rounds to $1$, though this is not important because you can move off the exact number by a small amount.

If Jack picks a number in $[0,0.05]$ or $[0.95,1)$ Colin will choose $1$ because they round to an integer. If Jack picks a number in $[0.475,0.525]$ Colin will choose $2$. If Jack picks a number in $[\frac 13-\frac 1{60},\frac 13+\frac 1{60}]$ or $[\frac 23-\frac 1{60},\frac 23+\frac 1{60}]$ Colin will choose $3$. Keep note of what of the interval has been covered as you go up through the choices for Colin. When the last bit of the interval is covered, you have the answer. $19$ seems reasonable because each number by itself covers a length of $0.1$ but there are areas of overlap.