Using matchsticks, we form digits as shown in the figure. In this way, to write the number 188, we use 16 matchsticks.

![](https://cdn.mathpix.com/cropped/2024_05_01_1c2853cb1112cfb2ee2dg-02.jpg?height=505&width=696&top_left_y=2183&top_left_x=589)

César wrote the largest number possible using exactly 13 matchsticks. What is the sum of the digits of the number that César wrote?
A) 8
B) 9
C) 11
D) 13
E) 15

#
Solution

## ALTERNATIVE B

A number with a certain number of digits, where the first digit from the left is not zero, is always greater than any number that has one less digit. For example, 1000 (with 4 digits) is greater than 999 (which has only 3 digits).

Thus, with exactly 13 sticks, we should form a number that has the maximum possible number of digits, with the first digit from the left being different from 0. Since, among all the digits, the digit 1 is the one formed with the fewest sticks, we see that to obtain the largest possible number with 13 sticks, we should use as many 1s as possible.

It is not possible to use the digit 1 six times, as in this case we would already have used 12 sticks and there is no digit that can be formed with only one stick. For the same reason, it is not possible to use the digit 1 five times; there is no digit formed by 3 sticks.

But it is possible to use the digit 1 four times; in this case, we use 8 sticks and can complete the number with one of the digits 2 or 5, which are formed by 5 sticks. In this case, we should choose the 5, which allows us to form the number 51111 with 13 sticks. The sum of the digits of this number is \(5+1+1+1+1=9\).

![](https://cdn.mathpix.com/cropped/2024_05_01_1c2853cb1112cfb2ee2dg-19.jpg?height=222&width=423&top_left_y=1508&top_left_x=722)