Write the numbers from 0 to 9 in the circles next to them, so that they increase in a counterclockwise direction. Then, subtract one unit from the odd numbers and add one unit to the even numbers. By choosing three consecutive circles, what is the highest sum you can obtain?
(a) 19
(b) 21
(c) 23
(d) 24
(e) 25

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-002.jpg?height=317&width=308&top_left_y=2394&top_left_x=1522)
The correct option is (c).

Starting from any circle, we initially get the sequence $0,1,2,3,4,5,6,7,8,9$. By subtracting one from the odd numbers and adding one to the even numbers, the sequence becomes $1,0,3,2,5,4,7,6,9,8$. It is now easy to verify that the largest possible sum of three consecutive numbers is $8+9+6=23$.
![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-037.jpg?height=398&width=1122&top_left_y=1154&top_left_x=500)