Answer to Question #270880 in Discrete Mathematics for kd bhai

Question

A patient is given a prescription of 45 tablets with the instructions to take at least one tablet per day

for 30 days. Prove that there must be a period of consecutive days during which the patient takes a

total of exactly 14 tablets.

Answer 1

Let X_i be the number of tablets the patient needs to taken within i^th day.

$1\leqslant X_1\leqslant X_2\leqslant X_3\leqslant..........X_{30}\leqslant45$

$X_1+14\leqslant X_2+14\leqslant X_3+14\leqslant..........X_{30}+14\leqslant45+14$

In this case, there are total 60 integers and 59 holes. It can say that 2 out of 60 integers must be the same. Since the X_i’s are distinct, so we have

X_i = X_j + 14

For the above i and j;

X_i - X_j = 14

So, its prove that;

X_j+1 + X_j+2 + ...........+ Xi = 14

Hence proved.