You have 2 glasses, each filled with 100 millilitres (ml) of soda. One contains cola and one contains orange soda. First, you take a spoonful of cola and mix it into the orange soda. Next, you take an equal spoonful of the orange-cola mixture and put it into the glass of cola.
Which glass contains the largest volume of the other soda? In other words: is there more cola in the orange soda or is there more orange soda in the cola?
Try keeping track of the different soda volumes for each glass like this: [cola, orange, total].
Next, pick a volume for the spoonfuls, say 25ml, and track the soda volumes in each glass through the problem.
Surprisingly, the glasses contain an equal volume of the other soda! In other words, there is the same amount of orange soda in the cola as there is cola in the orange soda.
First, a conceptual solution: let’s look at the glass of cola. At the start, it’s 100 ml, and at the end, it’s also 100 ml. It’s missing some cola. Making up the difference, there’s some orange soda in this glass of cola.
Since there’s still 100 ml of liquid in the glass, the amount of cola missing must equal the amount of extra orange in the cola bottle. And that equal amount of missing cola is currently in the glass of orange soda.
This means there’s exactly the same amount of orange soda in the cola glass as there is cola in the orange glass.
Don’t believe us? Here’s a second, more arithmetic solution:
Imagine that each spoonful is 25ml and that we can keep track of the different soda volumes for each glass using this format: [cola, orange, total].
At first the cola [100, 0, 100] and the orange [0, 100, 100] are pure sodas with no mixture.
After taking a 25ml spoonful of cola and mixing it into the orange soda, the orange soda is now [25, 100, 125] and the cola is [75, 0, 75].
Next, we take a 25ml spoonful from the orange-cola mixture [25, 100, 125]. How much of that spoonful is cola? Well, we know the mixture is 25ml cola out of 125ml total so 25/125 or 1/5. That means 1/5 of the spoonful will be cola. 1/5 of 25ml is 5ml cola. This means the spoonful is [5, 20, 25].
Taking this spoonful out of the orange-cola mixture makes this glass [25 – 5, 100 – 20, 125 – 25] or [20, 80, 100].
Adding that cola-orange spoonful to the cola makes the cola [75 + 5, 0 + 20, 75 + 25] or [80, 20, 100].
Both glasses have 80ml of the original soda and 20ml of the other soda!
Does it work with a different, bigger spoonful, say 60ml? It sure does:
Both glasses have 62.5ml of the original soda and 37.5ml of the other soda. This pattern will happen no matter what volume you choose for the spoonfuls. Pretty sweet!