Question

Two players A and B play the following game. A selects an integer from 1 to 10, inclusive of both. B then adds any positive integer from 1 to 10, both inclusive, to the number selected by A. The player who reaches 46 first wins the game. If the game is played properly, A may win the game if:

• A. A selects 8 to begin with
• B. A selects 2 to begin with
• C. A selects any number greater than 5
• D. None of the above

Hint:

Work backward from 46 in steps of 11 to find the checkpoint numbers 2, 13, 24, 35, 46, and see which starting number lets A control every checkpoint.

Answer 1

The Correct Option is B

Step 1: Understand the winning idea behind this type of game.
Each player can add any whole number from 1 to 10 on their turn.
Whenever it is your move and your opponent has just added some number \(m\) (where \(1\le m\le10\)), you can always add \(11-m\) in reply, because \(11-m\) is also between 1 and 10.
This means that after any full round (opponent's move plus your reply), the running total always goes up by exactly 11, and you get to fix that.

Step 2: Work backward from the target, 46, in steps of 11.
If you can always land the total on one of these "control" values, you can guarantee reaching 46 on your own move:
\[ 46,\ 46-11=35,\ 35-11=24,\ 24-11=13,\ 13-11=2 \]
So the control values, from first to last, are 2, 13, 24, 35 and 46.

Step 3: Apply this to who moves first.
Here A gets to pick the very first number outright, before B even plays.
So A should pick the very first control value, 2, as the opening number, to grab control of the game immediately.

Step 4: Check that this really forces a win.
Suppose A opens with 2. Whatever B adds, call it \(m\) (\(1\le m\le10\)), the total becomes \(2+m\).
A now adds \(11-m\), which is also a valid move (between 1 and 10), bringing the total to \(2+m+(11-m)=13\), the next control value.
The same idea repeats: A always answers B's move to reach 24, then 35, and finally 46 exactly on A's own turn, winning the game.

Step 5: Rule out the other options.
8 is not one of the control values 2, 13, 24, 35, 46, so if A opens with 8, B can grab control instead (for instance, by adding a number that lands on 13) and B, not A, ends up forcing the win.
"Any number greater than 5" is far too broad. Numbers like 6, 7, 9 or 10 are not control values, so most choices in this range do not guarantee A a win.
Since choosing 2 does guarantee a forced win for A, "None of the above" cannot be correct either.

Final Answer:
A should select 2 to begin with, to control the game and force a win at 46.
\[ \boxed{\text{A selects 2 to begin with}} \]