Minimum Information Sum
Alice has written on a piece of paper a sequence of numbers, and in that order. She keeps the paper hidden from Bob.
Alice tells Bob that there are exactly 5 numbers, written in order of magnitude, and the differences between adjacent terms are all the same.
Alice asks Bob to guess the sum of the numbers. She will allow Bob to ask for any of the numbers as a hint. What is the fewest number of hints Bob would need to correctly guess the sum?
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1 solution
If he knows that the difference between each consecutive term is the same and that there are five numbers, then he only needs the middle term as the sequence can be described like this: n - 2x, n - x, n, n + x, n + 2x, where n = 19. This simplifies to 5n, or 5 x 19 = 95.
If there were six numbers, again he would need the middle term, but this time it would be a decimal as the median of 6 numbers requires you to find the (6+1)/2 term i.e the 3.5th term.
(All the xs cancel out i.e. -2x +2x =0, -x + x =0)