Number Nightmare

What is the next number in the following sequence?

1   11   21   1112   3112   211213   ...

Factorial Fiasco

What is the next number in the following sequence?

0   1   2   720!   ...

Note that the '!' symbol refers to the factorial operation.

Quarter Queasiness

A friend of your proposes a new game. The playing board consists of a perfectly circular table. The game pieces are quarters, of which both of you have an infinite supply. The rules are as follows:

1. You may place a quarter anywhere on the table, but may not overlap other quarters.
2. If you cannot place a quarter on the table as per rule #1, you lose.

Your friend, being the gracious sort, asks if you would like to go first or second. Can you come up with a strategy that will beat him every time?

Resistor Riot

Imagine a perfect wire-frame cube (a cube with only edges and no faces). Each edge is actually a resistor with resistance R.

What is the resistance between point A and point B (in terms of R)?

Chain Challenge

Much to your surprise, you find that you have spontaneously become a vagabond, aimlessly walking the earth in search of high adventure. During your travels you manage to pick up a straight gold chain with exactly 7 circular links, and a very handy pair of gold link snippers. Contrary to popular belief, your snippers are more valuable to you than your chain, since gold is easily found, but snippers are much harder to come by in this pre-industrial world.

Your travels bring you to an inn, where you decide you are going to rest for 7 days. You ask the innkeeper how much payment she requires for your stay. She takes a fancy to your gold chain and informs you that it will cost you 1 gold link per day. Her terms are very simple:

At the end of each day of your stay, she must have in her possession the same number of gold links as the number of days that you have stayed at the inn.

You quickly figure out that this means she must have 1 gold link at the end of the first day, 2 gold links at the end of the second day, and so on.

Now, you are struck with a dilemma. There is no way to pay the innkeeper without using your gold link snippers. Unfortunately, overuse of the snippers could cause them to become dull prematurely, shortening their useful life. You also don't trust the innkeeper enough to pay her in advance for your stay.

With this in mind, you need to figure out the minimum number of cuts that you need to make in the gold chain to pay the innkeeper by her terms. How many cuts do you need to make in your gold chain, and why?

Lightbulb Lunacy

Your problem solving career takes you to a chemical plant in which very dangerous and downright noxious chemicals are manufactured and used. One day, there is a horrific leak and one of the rooms in the building is completely filled with a deadly gas. The only way in or out of the room is through a single door. There are no windows.

Unfortunately, this room happens to be the control room of the plant which contains the only emergency shutoff switch. There is only one oxygen mask left, since the others were all used up during the evacuation. This gas mask only has 2 minutes worth of oxygen in it. As a big stroke of misfortune, the emergency shutoff switch takes 3 minutes to properly operate.

Fortunately, however, the engineers at the plant have figured out how to bypass the shutoff circuitry. They are able to patch in all of the power conduits except the final three. These 3 conduits also happen to be hooked up to 3 bare incadescent light bulbs in the gassed room. The lightswitches are located outside of the room far away from the entrance.

The engineers tell you that if you can figure out which switch is connected to which light bulb, they can patch in the final circuits and shut down the plant before catastrophe ensues.

You realize that there is no way to look inside the room while throwing any particular switch, nor is there a way to communicate with anybody inside of the room. You can only place the switches in any configuration, walk inside the room with the last remaining gas mask, and walk out again within 2 minutes. Once spent, the gas mask cannot be used again. It dawns upon you that you only have one chance to get this right.

How can you safely determine which light switch is connected to which incandescent light bulb?

Mango Madness

Three mango merchants, known creatively as Al, Bob, and Charlie, each have different philosophies when it comes to selling their mangoes. Al believes in selling a low quantity of quality mangoes. Charlie believes in high volume mango sales, while Bob prefers moderation.

One day, they decided to put their selling philosophies to the test. They set down the following rules:

Rule 1: For the first half of the day, all three must sell their mangoes at the same price.

Rule 2: For the second half of the day, they must all lower their prices by the same amount and continue to sell their mangoes for the rest of the day.

Rule 3: They must all sell at least one mango at the higher price and one mango at the lower price.

So, satisfied with these rules, Al, Bob, and Charlie set up their shops with 10, 25, and 40 mangoes each, respectively. The day went on slowly, as competition was fierce, but finally the event drew to a close. The final judgement on their sales philosophies would now be made!

To their complete surprise, all three of them managed to not only sell all of their mangoes, but also end up with the same amount of money at the end of the day! What are all of the possible ways this can happen?