Hint: the function is decreasing at odd number of cups. and the function is constant at even number of cups. Look at this graph:

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Let's deal with the base case first. Let avg=(h+c)/2. Now temperature will never go below this. So if t<=avg, the answer is 2. Now, for all even no. of moves answer be average and for odd, the answer will tend towards avg. The temperature function after 'x+1' cups of hot water will be — ((x+1)*h + x*c )/(2*x+1). we can see that this tends to 'avg' as x tends to inf. Note here that the total moves are 2*x+1 which is always odd. Let's denote the total moves by 'n'(2*x+1). When n=1, temp is h, and as n increases, temp. keeps decreasing and tends to avg. (At n=inf, temp=avg).Now let's do binary search on the value of x. For some x, 't' will lie between temp at x and x+1. So low=0,hi=(some high value). We will calculate the value at mid and mid+1. I think further approach should be clear. If not, feel free to ask. Here is an implementation of above explained method:- 81782877