Rudolph gives Dasher and Dancer as many sleigh bells as each already has.
Then Dasher gives Rudolph and Dancer as many sleigh bells as each of them then has.
Finally, Dancer gives Dasher and Rudolph as many sleigh bells as each has.
If at the end, each has 16 sleigh bells, how many sleigh bells did each one have at the beginning?
Just read what it says, and write it down in symbols:
If we start out with
x = Rudolph's bells
y = Dasher's
z = Dancer's
after step 1 we have
x-y-z
2y
2z
After step 2 we have
2(x-y-z) = 2x-2y-2z
2y-(x-y-z+2z) = 3y-x-z
4z
After step 3 we have
4x-4y-4z
6y-2x-2z
4z-(2x-2y-2z+3y-x-z) = 7z-x-y
So, now just solve these three equations to get the original x,y,z
4x-4y-4z = 16
6y-2x-2z = 16
7z-x-y = 16
z=8
y = 14
x = 26
To solve this problem, it's helpful to work backwards.
Let's start with the end condition: each reindeer has 16 sleigh bells.
During the last step, Dancer gives Dasher and Rudolph as many sleigh bells as each has. This means that after this step, Dasher and Rudolph both have 16 + 16 = 32 sleigh bells each, and Dancer has 16 - 2*16 = -16 sleigh bells.
In the second-to-last step, Dasher gives Rudolph and Dancer as many sleigh bells as each of them then has. Since Dancer has a negative number of sleigh bells, Dasher can't give any to Dancer. Therefore, this step doesn't affect the number of sleigh bells Dasher and Rudolph have, so they both still have 32 sleigh bells each.
Finally, Rudolph gives Dasher and Dancer as many sleigh bells as each already has. Again, Dancer can't receive any sleigh bells since he has a negative number, so the only exchange happens between Rudolph and Dasher. Since they both have 32 sleigh bells, they will exchange 32 bells each, resulting in 32 - 32 = 0 sleigh bells for both Rudolph and Dasher.
Therefore, at the beginning, Rudolph, Dasher, and Dancer had 0 sleigh bells each.