Garage Sale

Find the binary of each number. Whichever number in binary form has three ones is the number you can get.

$12 = 1100_{2}$

$15 = 1111_{2}$

$19 = 10011_{2}$

$23 = 10111_{2}$

As you can see, 19 is the only number that has exactly three ones in its binary representation. Therefore, the answer is 19 .

As noted in another solution, $1 + 2 + 16 = 19$ , so it is possible to get the TV.

It is not possible to get the Oven (12): Since $12$ is even, $1$ cannot be included in the sum. Neither can $16$ as it is too large. The only remaining option is $2 + 4 + 8 = 14$ , which will not win the Oven.

It is not possible to get the AC (15): The only way to get an odd sum is to include the number $1$ in your sum. Again, $16$ cannot be used as it is too large. This leaves only $1 + 2 + 4 = 7$ and $1 + 2 + 8 = 11$ , neither of which wins the AC.

It is not possible to get the Fridge: Again, we must include $1$ in order to get an odd sum. As seen in the previous step, we cannot get a sum of $23$ if we exclude $16$ from our sum. So $16$ must also be in our sum. We've also seen that $1 + 2 + 16 = 19$ wins the TV, not the Fridge. This leaves only $1 + 4 + 16 = 21$ and $1 + 8 + 16 = 25$ , neither of which wins the Fridge.

1+2+16 = 19. That will get you the TV19.