Brainf***, 61 bytes

Chinese, 16 characters

This requires the input is in the ASCII range 1-127 and is null-terminated. It deletes pairs of characters from the start and end of the string until there are one or two characters remaining. If there are two, it adds them together, then divides by 2, rounding down. The remaining character is printed.

,[>,]<<<[[<]>[-]>[>]<[-]<<<]>[[->+<]>[-[-<+<]>>]<[>]<[>]<<]>.

Try it on this interpreter.

Dissection:

,[>,] Get the null terminated input
<<< Move to the second last character
[ While there are at least 3 characters
 [<]> Move to the first character
 [-] Delete it
 >[>]< Move to the last character
 [-] Delete it
 <<< Move the the third last character
]
> Move to the second last character
[ If there are two characters remaining
 [->+<] Add the two characters together
 >[-[-<+<]>>] Divide the character by 2 rounding down
 <[>]<[>]<< Move to before the character to exit the if loop
]
>. Print the remaining character

* Given each instruction could be compressed to 3 bits and encoded in UTF-32, the whole program could technically be expressed in 6 characters.

EDIT: And thanks to Jan Dvorak for introducing me to Chinese, which compresses this into 16 characters, on par with Dennis' CJam answer.

蜐蕈帑聿纂胯箩悚衅鹊颂鹛拮拮氰人

Brainf***, 61 bytes

Chinese, 16 characters

This requires the input is in the ASCII range 1-127 and is null-terminated. It deletes pairs of characters from the start and end of the string until there are one or two characters remaining. If there are two, it adds them together, then divides by 2, rounding down. The remaining character is printed.

,[>,]<<<[[<]>[-]>[>]<[-]<<<]>[[->+<]>[-[-<+<]>>]<[>]<[>]<<]>.

Try it on this interpreter.

Dissection:

,[>,] Get the null terminated input
<<< Move to the second last character
[ While there are at least 3 characters
 [<]> Move to the first character
 [-] Delete it
 >[>]< Move to the last character
 [-] Delete it
 <<< Move the the third last character
]
> Move to the second last character
[ If there are two characters remaining
 [->+<] Add the two characters together
 >[-[-<+<]>>] Divide the character by 2 rounding down
 <[>]<[>]<< Move to before the character to exit the if loop
]
>. Print the remaining character

* Given each instruction could be compressed to 3 bits and encoded in UTF-32, the whole program could technically be expressed in 6 characters.

EDIT: And thanks to Jan Dvorak for introducing me to Chinese, which compresses this into 16 characters, on par with Dennis' CJam answer.

蜐蕈帑聿纂胯箩悚衅鹊颂鹛拮拮氰人

Brainf***, 6 characters*, 61 bytes

This requires the input is in the ASCII range 1-127 and is null-terminated. It deletes pairs of characters from the start and end of the string until there are one or two characters remaining. If there are two, it adds them together, then divides by 2, rounding down. The remaining character is printed.

,[>,]<<<[[<]>[-]>[>]<[-]<<<]>[[->+<]>[-[-<+<]>>]<[>]<[>]<<]>.

Try it on this interpreter.

Dissection:

,[>,] Get the null terminated input
<<< Move to the second last character
[ While there are at least 3 characters
 [<]> Move to the first character
 [-] Delete it
 >[>]< Move to the last character
 [-] Delete it
 <<< Move the the third last character
]
> Move to the second last character
[ If there are two characters remaining
 [->+<] Add the two characters together
 >[-[-<+<]>>] Divide the character by 2 rounding down
 <[>]<[>]<< Move to before the character to exit the if loop
]
>. Print the remaining character

* Given each instruction could be compressed to 3 bits and encoded in UTF-32, the whole program could technically be expressed in 6 characters.

Brainf***, 61 bytes

Chinese , 16 characters

This requires the input is in the ASCII range 1-127 and is null-terminated. It deletes pairs of characters from the start and end of the string until there are one or two characters remaining. If there are two, it adds them together, then divides by 2, rounding down. The remaining character is printed.

,[>,]<<<[[<]>[-]>[>]<[-]<<<]>[[->+<]>[-[-<+<]>>]<[>]<[>]<<]>.

Try it on this interpreter.

Dissection:

,[>,] Get the null terminated input
<<< Move to the second last character
[ While there are at least 3 characters
 [<]> Move to the first character
 [-] Delete it
 >[>]< Move to the last character
 [-] Delete it
 <<< Move the the third last character
]
> Move to the second last character
[ If there are two characters remaining
 [->+<] Add the two characters together
 >[-[-<+<]>>] Divide the character by 2 rounding down
 <[>]<[>]<< Move to before the character to exit the if loop
]
>. Print the remaining character

* Given each instruction could be compressed to 3 bits and encoded in UTF-32, the whole program could technically be expressed in 6 characters.

EDIT: And thanks to Jan Dvorak for introducing me to Chinese , which compresses this into 16 characters, on par with Dennis' CJam answer .

蜐蕈帑聿纂胯箩悚衅鹊颂鹛拮拮氰人

Brainf***, 3 characters*, 61 bytes

This requires the input is in the ASCII range 1-127 and is null-terminated. It deletes pairs of characters from the start and end of the string until there are one or two characters remaining. If there are two, it adds them together, then divides by 2, rounding down. The remaining character is printed.

,[>,]<<<[[<]>[-]>[>]<[-]<<<]>[[->+<]>[-[-<+<]>>]<[>]<[>]<<]>.

Try it on this interpreter.

Dissection:

,[>,] Get the null terminated input
<<< Move to the second last character
[ While there are at least 3 characters
 [<]> Move to the first character
 [-] Delete it
 >[>]< Move to the last character
 [-] Delete it
 <<< Move the the third last character
]
> Move to the second last character
[ If there are two characters remaining
 [->+<] Add the two characters together
 >[-[-<+<]>>] Divide the character by 2 rounding down
 <[>]<[>]<< Move to before the character to exit the if loop
]
>. Print the remaining character

* Given each instruction could be compressed to 3 bits and encoded in UTF64, the whole program could technically be expressed in 3 characters.

Brainf***, 6 characters*, 61 bytes

This requires the input is in the ASCII range 1-127 and is null-terminated. It deletes pairs of characters from the start and end of the string until there are one or two characters remaining. If there are two, it adds them together, then divides by 2, rounding down. The remaining character is printed.

,[>,]<<<[[<]>[-]>[>]<[-]<<<]>[[->+<]>[-[-<+<]>>]<[>]<[>]<<]>.

Try it on this interpreter.

Dissection:

,[>,] Get the null terminated input
<<< Move to the second last character
[ While there are at least 3 characters
 [<]> Move to the first character
 [-] Delete it
 >[>]< Move to the last character
 [-] Delete it
 <<< Move the the third last character
]
> Move to the second last character
[ If there are two characters remaining
 [->+<] Add the two characters together
 >[-[-<+<]>>] Divide the character by 2 rounding down
 <[>]<[>]<< Move to before the character to exit the if loop
]
>. Print the remaining character

* Given each instruction could be compressed to 3 bits and encoded in UTF-32, the whole program could technically be expressed in 6 characters.