Western Leaders

Here's a freaky coincidence about World War II. If you add up the year of birth, age in 1944, year of taking power, and the number of years in office in 1944 for each of the five main leaders of the Western world during World War II, the sums are all the same.

	Churchill	Hitler	Mussolini	Roosevelt	Stalin
Year of birth	1874	1889	1883	1882	1878
Age in 1944	70	55	61	62	66
Took power	1940	1933	1922	1933	1922
Years in office	4	11	22	11	22
Sum	3,888	3,888	3,888	3,888	3,888

Can you explain this coincidence? Click below for the answer.

There's really no coincidence at all. If you take any person's year of birth and add their age in 1944, the sum will be... 1944. This also goes for the year a leader took office and the number of years they had held office in 1944, so the sum of all four values will always be 1944 + 1944 = 3888. You can make your own puzzle using any year as a reference point.

Concentric Shapes

See the image below of a square inscribed inside a circle inscribed inside a square. If the outer square has an area of 100 square inches, is there a quick way of figuring out the area of the inner square? Click below for the answer.

Spin the inner square around so the corners touch the top, bottom, and sides of the outer square.

Now imagine taking each corner of the outer square and folding them in so they touch in the center of the image. It's easy to see that the area outside the inner square will perfectly cover the area inside, so the inner square has exactly half the area of the outer square, or 50 square inches.

Digit Frequency

If you write down all the numbers from 1 to 1000 (inclusive) which digit occurs most frequently? Which digit appears least frequently in the same range? Click below for the answers.

You could write all the numbers down or write a program to count the digit frequencies, but this problem is much easier if you look for a pattern. The digits 0 through 9 all appear the same number of times in the ones place, but 1 through 9 appear a lot more often in the tens and hundreds place (because of numbers like 22, 222, etc., and the fact that we don't write leading 0s). This means that 0 is the least frequent digit. It only appears 192 times in the specified range. What about the most frequent? The digits 1 through 9 appear with exactly the same frequency in the range 1 to 999 (300 times each), so that extra occurrence (in the number 1000) makes 1 the most frequent digit from 1 to 1000.

Counting Chickens

If one-and-a-half chickens lay one-and-a-half eggs in one-and-a-half days, how many eggs does one chicken lay in one day? Click below for the answer.

For many people, the intuitive answer is one egg, but it pays double-check your math on this kind of problem. The daily rate of eggs per chicken per day is given by the formula

daily rate = eggs / (chickens x days)

Plugging in the numbers from the first part of the problem, we get

daily rate = 1.5 / (1.5 x 1.5)
daily rate = 1.5 / 2.25
daily rate = 2/3

So, one chicken lays two-thirds of an egg in one day.

Number Sense

How good is your "number sense"? How many of the following can you answer without using a calculator or looking up a conversion factor?

1. Are there more inches in a mile, or Sundays in 1000 years?
2. Are there more seconds in a week, or feet in 100 miles?
3. Are there more millimeters in a mile, or seconds in a month?
4. Which is larger, multiplying all the numbers from 1 to 10, or multiplying just the even numbers from 1 to 16?
5. Which is longer, 666 days or 95 weeks?
6. Which is longer, 666 inches or 55 feet?
7. Which is longer, 666 hours or 28 days?
8. Are there more ounces in a ton or inches in a kilometer?
9. Which is hotter, $0^{\circ}C$ or $0^{\circ}F$?
10. Which is larger, $e^\pi$ or $\pi^e$?

Click below for the answers.

1. Inches in a mile. (63,360. There can be up to 52,178 Sundays in 1000 years.)
2. Seconds in a week. (604,800, compared to 528,000 feet in 100 miles.)
3. Seconds in a month. (Even if the month only has 28 days, that's 2,419,200 seconds, compared to only 1,609,340 millimeters in a mile.)
4. Just the even numbers from 1 to 16. (Multiplying all the numbers from 1 to 10 gives you 3,628,800. Multiplying the even numbers from 1 to 16 give you 10,321,920.)
5. 666 days. (95 weeks is only 665 days.)
6. 666 inches. (55 feet is 660 inches.)
7. 28 days (which is 672 hours).
8. Inches in a kilometer. (39,370.1, compared to 35,840 ounces in a long ton, which is the heaviest ton.)
9. $0^{\circ}C$ is "hotter" since it is equal to $32^{\circ}F$
10. $e^\pi$ (23.14) is larger than $\pi^e$ (22.46).