## Blue eyed cows brainteaser

By David, 3 April 2019 Brainteaser

Imagine a place called Occhiblu station, where they’re trying to breed cows with blue eyes. They have 100 cows on the station, and some of them have one blue eye. Of the rest of the cows, half have two blue eyes.

Recently, one of the farmhands noticed that blue eyes are sensitive to light, so they want to put shaded contacts on all the blue eyes in the herd.

How many blue eyes are there?

Scroll down for the answer!

## Brainteaser answer

In this puzzle, we’re interested in three types of cows. Some have one blue eye, some two, and some none.

How many cows have no blue eyes? Ignoring the ones with one blue eye, we know that half have two blue eyes. The remaining half must have no blue eyes!

Now we can pair up the cows. Each cow with two blue eyes can pair with a not blue eyed cow. And each one blue eyed cow can pair with another one blue eyed cow. Each pair has two cows and two blue eyes.

With 100 cows there are 50 pairs, so there are 100 blue eyes.

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6 April, 2019 at 2:12 pm

Cute little problem, but the pairing argument seems a bit unintuitive. Clearer to just prove it, and practise formulating at the same time:

Let E be the number of blue eyes.

Let B1 be the number of cows with one blue eye.

Let B2 be the number of cows with two blue eyes, which is half of the remainder, so

B1 + 2*B2 = 100, so B2 = (100 – B1)/ 2

E is the number of one-eyed cows plus twice the number of two-eyed cows, or

E = B1 + 2 * B2, and we already know that B1 + 2*B2 = 100. Or we can work it through:

B1 + 2*B2 = B1 + (2 * (100 – B1) / 2) = B1 + 100 – B1 = 100 … Q.E.D. 🙂