## Saturday, August 19, 2017

### Factor Sums

Not counting itself, the number 6 has the factors 1, 2, and 3, which add to 6. The number 28 has the same property (its factors are 1, 2, 4, 7, and 14). Can you come up with a three-digit number that has this property? What about a four-digit number? Click below for the answers.

## Saturday, August 12, 2017

### Minimum Percentage

75% of men from a certain group are tall, 75% have brown hair, and 75% have brown eyes. What is the minimum percentage that are tall, have brown hair,

*and*have brown eyes? Click below to see the answer.

Instead of thinking in percentages to solve this problem, it's helpful to think back to the Pigeonhole Principle. Think of a group of 100 men, then 75 are tall, 75 have brown hair, and 75 have brown eyes. That's 225 individual attributes to assign to 100 men, so at least 25 of them (or

**25%**) must have each of the three attributes.## Saturday, August 5, 2017

### 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

$36 = 2 * 3 * 6$

$10x + y = 2xy$

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

$36 = 2 * 3 * 6$

## Saturday, July 29, 2017

### 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.)

## Saturday, July 22, 2017

### 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.

Labels:
logic puzzles

## Saturday, July 15, 2017

### 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

Labels:
logic puzzles

## Saturday, July 8, 2017

### 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.)

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