A Two-Digit Number

Find a two-digit number that's equal to two times the result of multiplying its digits. Click below to see the answer.

My first attempt at solving this puzzle was to set it up as an equation and try to solve it algebraically. Let's say the two digits are $x$ and $y$. Then the equation would be:

$10x + y = 2xy$

The left-hand side is the two-digit number ($x$ in the tens place, $y$ in the ones place) and the right-hand side is two times the result of multiplying its digits. If you try to isolate either $x$ or $y$, you'll see that it's not very easy to come up with a clean solution. That's because the equation above describes a hyperbola.

That's not exactly a dead end, but it isn't the kind of easy-to-understand (once you see it) solution I like in a logic puzzle. Luckily, there's an easier way. There aren't that many possibilities (we're only dealing with two digits), and we can eliminate a lot of them.

For example, we know that neither digit is 0. Also, we know that $2xy$ is an even number, so $y$ must be even (because the result of adding it to an even number is even). We also know that the product of the digits must be less than 50, otherwise $2xy$ would have three digits. That gets us down to only 32 possibilities to test.

Any other shortcuts that I can think of would only eliminate a few possibilities, but it's easy to just test the remaining ones (I went through them manually, but you could write a short script or use a spreadsheet), and find that the solution is

$36 = 2 * 3 * 6$

Identical Twins

Alice and Eve are identical twin sisters. One always lies and the other always tells the truth, but we don't know which is which. We ask one of them "Is Alice the one that always lies?" and she replies "Yes." Did we speak to Alice or Eve? Click below to see the answer.

We spoke to Eve. A person who always lies or always tells the truth cannot admit that they are a liar, so Alice could not have answered "Yes" to that question. (Note that we still don't know which sister is the liar and which is the truth teller.)

Counting Socks

All my socks are red except two. All my socks are white except two. All my socks are blue except two. How many socks do I have? Click below for the answer.

Oddly, I only have three socks. Don't let the fact that socks normally come in matching pairs distract you. No other number satisfies all three conditions above.

Circumnavigation

From 1519 until 1522, Ferdinand Magellan's Victoria was the first ship to successfully circumnavigate the globe. (Magellan himself did not survive the entire voyage.) Can you tell me which part of the ship traveled the greatest distance? Click below for the answer.

If you remembered the Rope Around the Earth puzzle I posted a few months ago, you probably got this one pretty quickly. Since the world is roughly spherical, the tip of the tallest mast of the ship would have traveled the greatest distance in sailing around the globe. Imagine if a boat sailed in a perfect circle around the equator. The part of the boat deepest under water (the keel) would create a smaller circle than the tip of the mast several feet above the water, so the tip of the mast travels the greatest distance during the voyage.

Replica of the Victoria, Photograph by Gnsin - Own work, CC BY-SA 3.0

50 factorial

50! = 30414093201713378043612608166064768844377641568960512071337804000

Without doing the full computation, can you tell whether the above statement is true or false? Click below for the answer.

You can probably guess that the statement is false, otherwise it wouldn't be much of a puzzle. The reasoning, though, is that the factorial for 50 must include the factors 10, 20, 30, 40, and 50, so it must end in at least five zeroes. The value above ends in only three zeroes, so it cannot be correct. (The correct value is 30414093201713378043612608166064768844377641568960512000000000000.)

The Missing Fish

Two fathers took their sons fishing. Each man and his son caught one fish, but when they all returned to camp they only had three fish. None of the fish were eaten, lost, or thrown back. How could this be? Click below to see the answer.

There were only three people on the fishing trip. One man was the father and grandfather of the other two.

Bags of Marbles

You have three identical bags, each containing two marbles. Bag A contains two white marbles, Bag B contains two black marbles, and Bag C contains one white and one black marble. You pick a bag at random and draw out one marble. If the marble is white, what is the probability that the other marble in the same bag is also white? Click below to see the answer.

Many people will instinctively answer 50%, or 1/2, since the marble has two possible colors, but the probability is actually 2/3 (66.67%). Why? If the first marble is white, then you know you didn't randomly select Bag B. That means that the first marble you selected has three (not two) possibilities:

1. The first marble in Bag A.
2. The second marble in Bag A.
3. The white marble in Bag C.

Only in the third case will the other marble be black, so there's a 2/3 probability that when the first marble is white, the second marble will also be white.

If you want to see how you would model this problem in Python, you can look at my solution on GitHub.