Game Booths

4

Referring to the above diagram,

Z + 80 + Y + 8 = 1/2 of area of ABCD

? + Z + 72 + Y + 12 = 1/2 of area of ABCD

Therefore, Z + 80 + Y + 8 = ? + Z + 72 + Y + 12.

? = 4

The likeliness of a baby being a girl or a boy is the same. So the answer is 1:1!

If this simple ratio doesn't convince you, here's a more complicated method.

Suppose there are N child-bearing couples.

For the first baby born, half will be girls, half will be boys. The half with firstborn baby girls (N/2) will only have 1 child, since they should stop bearing children after having a baby girl. The half with firstborn baby boys (N/2) will give birth to the second baby.

Half of the those with firstborn baby boys (N/4), will only have 2 children: one boy and one girl, since they should stop bearing children after having a baby girl. The other half, with 2 baby boys (N/4), will give birth to the third baby.

Half of those with 2 baby boys (N/8) will only have 3 children: two boys and one girl, while the other half (N/8), with 3 baby boys will give birth to the fourth baby. And so on...

The total number of children can be found from this infinte geometric series:

N + N/2 + N/4 + N/8 + N/16 + ...

The sum of the series is 2N. Since there will be exaclty 1 girl from each couple, there will be N girls at last, which is 50% of all children. Ratio of girls to boys is 1:1.

The solution here is amazing if you pick the right strategy. Just to set the scene, the simplest strategy would be to just switch each circuit breaker off one at a time. But this would take 99 trips to the basement—the 100th circuit breaker would be mapped by the process of the elimination). You can do much, much better.

Believe it or not, you can map all 100 circuit breakers to their respective lights in just 7 trips to the basement!

Here’s the strategy:

For the ease of keeping track of things, put a piece of masking tape on each circuit breaker and on each light. On the first trip to the basement, flip 50 circuit breakers to off, mark these circuit breakers with a “0,” and mark the 50 circuit breakers that are on with a “1.” Accordingly, as you roam around the house to tally the lights, mark the 50 lights that are off with a “0” and mark the other 50 lights with a “1.”

On your second trip to the basement, keep off half of the circuit breakers that are marked with a “0,” turn off half of the circuit breakers that are marked with a “1,” and mark all of these circuit breakers with a second number of “0.” Flip on all other circuit breakers if they’re not already on, and mark their second number as “1.” Now go around the house, and again mark the lights that are off with a “0” and those lights that are on with a “1.”

At the end of this step, all of your circuit breakers and lights should be marked with either “00,” “11,” “10,” or “01.” In fact, you’ve completely separated the matching problem into four different groups of 25—i.e., all lights must be matched to a circuit breaker with their same two-digit code.

You’ll continue this process: In the third trip, flip half (or actually, 13 since 25 is an odd number) of all of the circuit breakers in each group ( “00,” “11,” “10,” and “01”) to off, and mark them with an additional “0.” Mark the 12 “on” circuit breakers in each group with a “1.” Go around the house, and once again mark all lights that are off with a “0” and all lights that are on with a “1.”

Now, you’ll have created eight different groups of either 12 or 13 lights and circuit breakers: “000,” “100,” “011,” “111,” “010,” “110,” “001,” and “111.” The lights still must be matched to a circuit breaker with the same three-digit string.

After your fourth trip, you’ll have subdivided the groups into 16 groups with either six or seven lights and circuit breakers in each group. After the fifth trip, you’ll have 32 groups with three or four lights and circuit breakers in each. And after the sixth trip, you’ll have either one or two lights or circuit breakers in each group.

For those groups that each have one light and one circuit breaker, you’ve successfully mapped those circuit breakers to their lights! For the rest, it takes only one more trip—the seventh trip—to finally map them to their respective lights.

If you’re familiar with binary numbers, you’ll recognize that there are exactly 128 numbers that use seven digits, 0000000 to 1111111, which may help to explain why this strategy works so well: Each circuit breaker and light ends up with a unique “code” that maps to a specific binary number.

Using this strategy, eight trips would allow you to map up to 256 circuit breakers, nine trips would get you to 512, and 10 trips would map up to 1,024!