##### Riddle 1

The key to this math riddle is realizing that the one place must be zero. 888 + 88 + 8 + 8 + 8 = 1,000.

Logic Games And Riddles

Other Math Brain Teasers:

One of the 'fathers' is also a grandfather. Therefore the other father is both a son and a father to the grandson.

In other words, the one father is both a son and a father.

The most common digit is '1.' Can you figure out why? No hints until you try the next riddle because the next riddle is closely tied to this one.

**Part II.** What digit is the least frequent between the numbers 1 and 1,000?

0 is the least common digit even though 1,000 has three zero's!

**Explanations for both riddles**

The digits 0 through 9 all follow the same pattern there is exactly 1 occurrence of each digit for every ten numbers.

- For instance the digit 2 appears once between 10 and 19, at 12. And 2 appears once between, 30 and 39 at 32.
- However, each of the digits 1 through 9 also appear in other numbers in the tens and hundreds place.

Again, let's look at 2 which appears in 20, 21, 22, 23, etc.. as well as 200, 201, 202, 203.

So to figure out how to answer the first riddle you had to see what distinguishes the number 1? Only that we are including 1,000 which would be the first '1' in a new series of ten! In other words, the digit 1 only has a single extra occurrence (301 occurrences) compared to 2 or 3 or 9 which each have exactly 300 occurrences.

The reason that zero has the least (BY FAR at only 192 occurrences) is because zero does not have any equivalents to 22, 33, 44, 222, 3333 etc.

There are many ways of explaining/thinking about this truly brain bending riddle! It all boils down to the fact that the lawyers's math is incorrect.

They did NOT spend $$\$9$$ • 3 + $$\$2$$.

They spent exactly $$\$27$$ dollars. $$\$25$$ for the room and $$\$2$$ for the tip. Remember they got exactly $$\$3$$, in total back.

Another way to think about the answer to this riddle is to just pretend that the bellhop refunded $$\$3$$ to the lawyers (rather than giving them $$\$5$$ and receiving $$\$2$$ back).

If the lawyers get $$\$3$$ back and each takes $$\$1$$. Then they spent exactly $$\$27$$ dollars.

The answer is 4.

7 large boxes (7 * 8 = 56 boxes)

4 small boxes (4 10 = 40 boxes

11 total cartons and 96 boxes

The key to solving this riddle is realizing that you have to take the rabbit over first and the switch the fox with the rabbit. See step 2.

Step 1Take the rabbit to the other side.

Shore |
Other Side |

Carrots Fox |
Rabbit |

Step 2

Go back and get the Fox and switch it with the Rabbit.

**The key here is that the carrots and the rabbit are not being left alone.

Shore |
Other Side |

Carrots Rabbit (Not left alone) |
Fox |

Step 3

Take the carrots across.

Shore |
Other Side |

Rabbit | Fox Carrots |

Step 4

Go back and get the rabbit.

Shore |
Other Side |

Rabbit Fox Carrots |

Every farmer's part is $$1/3(45+75) = 40$$ sacks.

Charlie paid $$\$1400$$ for $$40$$ sacks, then 1 sack costs $$\$1400/40 = $35/{\text{sack}}$$.

Adam got $$\$35*(45-40)=35*5 = \$175$$.

Ben got $$\$35*(75-40)=35*35 = \$1225$$.

**Answer:** Ben $$\$1225$$, Adam $$\$175$$

Why would he need to go back to get the last hint after seeing the number on the house next door?

Answer:

{2, 2, 9}

If you list out the trio of factors that multiply to 36 and their sums, you get:

- 1 1 36 = 38
- 1 2 18 = 21
- 1 3 12 = 16
- 1 4 9 = 14
- 6 6 1 = 13
- 2 2 9 = 13
- 2 3 6 = 11
- 3 3 4 = 10

Since the number on the house next door is not enough information there must be more than 1 factor trio that sums up to it, leaving two possibilities: { 6, 6, 1} , {2, 2, 9} . When she says her 'oldest' you know it can *not* be {6, 6, 1} since she would have two 'older' sons not an 'oldest'.

*change your choice*. If you don't believe me, just try out our free online Monty hall simulation.

Look at how many *closed areas* there are.

- 9999 has 4 closed areas (the top of the '9').
- 8888 has 8 closed areas, the top and bottom parts of the 8 and there are no other digits.
- 1816 has 3 closed areas, (top and bottom of 8 and bottom of 6, and it has 2 other digits (3 * 2 = 6)).
- 1212 has 0 closed areas, (0 * 4 = 0).

She will **never** get to the 1 foot mark because you keep adding smaller and smaller amounts!

Zeno's paradox of Achilles and the Tortoise

Quick summary of riddle: Zeno of Elea (490-425 BC) is known for creating many paradoxes which were debated by mathematicians for centuries. His riddle involving Achilles, the character from Homer's *Iliad* and a tortoise went something like:

The tortoise challenged Achilles to a race and Achilles, full of typical hubris, accepted and even gave the Tortoise a 10 foot head start. Before the race started, the tortoise told Achilles that the reason Achilles would lose is that even though Achilles would be catching up, the tortoise would always be moving ahead. Therefore, Achilles would always be covering a fraction of the distance between the two. Let's say he covered half of the distance in 1 second (5 feet) and then in the next he covered half of the **new **distance, the remaining 5 feet plus the Tortoise's new distance. In the end, the tortoise convinced Achilles that he could not win the race because although he would be getting closer and closer, he would still always be covering smaller and smaller fractions of the total distance between the two. Therefore, Achilles forfeited the race. To read this riddle in a modern narrative form click here.

Other Math Brain Teasers: