Don’t be so negative

This week’s Car Talk Puzzler is as follows:

In a certain family each daughter has the same number of brothers and sisters. Each son has twice as many sisters as brothers. How many sons and daughters are there in the family?

This is obviously a very straightforward problem in algebra. Let G be the number of girls in the family and B be the number of boys. Each girl has G-1 sisters and B brothers. Each boy has G sisters and B-1 brothers.

Converting the problem statement to a system of equations gives us:

G-1 = B
2G = B – 1

Now it’s a simple matter to rearrange the terms and solve for G or B. Let’s start by adding 1 to each side of the second equation:

2G + 1 = B

Now let’s substitute G – 1 for B, since the first equation tells us they’re equal:

2G + 1 = G – 1

Now let’s add 1 to each side again.

2G + 2 = G

Now let’s subtract G from each side.

G + 2 = 0

Finally, let’s subtract 2 from each side.

G = -2

…which implies that B = -3. So this particular family has negative two daughters and negative three sons.

Either the family is composed of antimatter, or my ability to do really simple algebra is gone, or there’s some subtle trickery in the wording of this puzzle that I’m missing. For example, maybe the trick is hiding in the difference between the words “and” and “as”: “each daughter has the same numbers of brothers and sisters,” while “each son has twice as many sisters as brothers.” The first sentence could be parsed to mean, “Each daughter has the same number of brothers-and-sisters (i.e., siblings) as the other daughters do,” which is trivially true no matter how many sons and daughters there are. But that reduces the system of equations to just:

2G = B – 1

which has infinitely many solutions for G and B. So where’s the error?

Update: I’m an idiot, as pointed out (gently) in the comments. “Each son has twice as many sisters as brothers” doesn’t mean

2G = B – 1

it means

G = 2(B – 1)

So the family is made of ordinary matter after all.

2 Replies to “Don’t be so negative”

  1. Each son has twice as many sisters as brothers.

    G = 2* (B-1)

    Thus

    G= 4
    B=3

    (sons have 2 brothers and 4 sisters, Daughters have 3 sisters and 3 brothers)

  2. Pick a specific number of sisters and brothers that satisfy your intuitive understanding of the English phrase “twice as many sisters as brothers.” For example, five brothers and ten sisters would be “twice as many sisters as brothers,” right?

    Now check that against your translation of that phrase into math. If G = 10 and B – 1 = 5, does 2G = B – 1?

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