Ten Problems

I have 61 identical looking coins in my pocket. Two of the coins are counterfeit, and the other 59 are solid gold. The only difference between the counterfeit and the gold coins is their weight, but we don’t know which one is heavier. How can I use a balance to find out whether a counterfeit coins or a gold coin is heavier?

Solution:

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There are many more ways to solve this problem. Can you figure them out?

Generalizing the problem to a total of 6k+1 coins (k a positive integer) including two counterfeits, can you find a general method?

A truck has two front tires that need to be changed every 15 000 km and four back tires that need to be changed every 25 000 km. (The difference is due to the imbalance of the load in the truck.) How can you switch the front and back tires to maximize the distance travelled without getting any new tires? What is that distance?

The Maximum distance is 20454+6/11 km.

Let us call the amount of ‘usable’ rubber in each tire a ‘rubber unit’, so we have six rubber units in total. From the information we are given, for every 1 km travelled, each front tire uses up 1/15000 rubber units and each back tire uses up 1/25000 rubber units. So, for every 1 km travelled, 2/15000 + 4/25000 = 11/37500 rubber units are used. Let us assume that the truck travels x km, so 11x/37500 rubber units are used. Since the maximum number of rubber units we have is 6, therefore 11x/37500 is less than or equal to 6, and x is less than or equal to 20454+6/11 km.

While we are traversing through this distance, we stop every third of the way, and switch two of the back tire with the front. In this case, each of the six tires would be installed at the front for 1/3 of the way and at the back 2/3 of the way. This way, upon arrive at our destination, all six tires will be used up.

In a catalog, you read about a set of blocks. There are 1029 blocks; all are identical in volume. They can be assembled into several tiers which are 1 foot thick and stack to form a pyramid. (The pyramid has a square base; its four sides are equivalent isosceles triangles.) How tall is this pyramid?

You have a three inch by three inch cube whose outside surface is painted blue. If you cut the cube into smaller cubes each with 1 inch dimensions, and you throw the new smaller cubes onto a mat, what’s the probability that the top surface of every cube will be blue?

For which number can you move the ones digit to the front and  in effect doubling the original number? (i.e. ABC becomes CAB, and )

Your assignment is to make a delivery from the depot to your base camp, 1600 miles away. The trip begins normally, but unfortunately, halfway to base camp, your supply truck breaks down. You have no way to call for help. Luckily, in addition to medical supplies, the truck is carrying a Desert Patrol Vehicle (DPV) and two barrels of fuel.

The DPV has a full 10 gallon tank, but to make room for the medical supplies, it can only carry one of the 45 gallon barrels at a time. While you can’t transfer fuel between barrels, you can refill the tank from the barrels. Assume the DPV can get 12 miles per gallon, regardless of the load it carries.

How far can you go? Is it far enough to deliver the medical supplies and the DPV to your base camp?

Solution

What is the next member of the following odd sequence:

663, 896, 84733, 3687, ?

Solution

Have you heard of the Four Squares Theorem? It states that every non-negative integer can be expressed as the sum of four squares.

Complete the following equality:

SQUARE + SQUARE + SQUARE + SQUARE = NUMBER

Replace the following letters with digits to satisfy the equation.

SPAM+SPAM+SPAM+SPAM+SPAM+SPAM=EMAIL

There is a hat with three pancakes in it: One is golden on both sides, one is brown on both sides, and the third is golden on one side and brown on the other. You take out one pancake, look at only one side, and observer that it is brown. What is the probability that the other side is also brown?

The last three digits of Allison’s license plate have a product of 360. The sum of the digits is 22 and the digits are in order from the least to greatest. What are the three digits of her license plate?

In a game of Swaff, players have 1 turn to roll 2 dice, to achieve a total number that is a multiple of 3. If the combined number is NOT a multiple of three, the player is out. If it is, he/she goes on to the next round. If no players get a multiple of 3, in any round, the game is over in a tie.

Example: John rolls a 2 and a 4. Total score is 6 - a multiple of 3. John would then move on to the next round.

Lets say that John, Sally, Suzan, and Jimmy were playing a game of Swaff. What is the probability that the game ends the first round as a tie?

Here’s a fun game for two. The first person says a number between 1 and 10. Then the other will say a number that is at least 1 higher and at most 10 higher than the previous number. They go back and forth until one says 50 and wins. What’s the winning strategy for the person who starts?

Since Percy posted the 1=2 thing a few days ago, I’m tempted to make up something of my own. So here it is…

To proof:

proof:

multiply both sides by , we have

From property of logarithms, we have

Therefore, , and

What went wrong?

Update: Solutions are posted!

Tony and Janet decide to play a game to settle their differences. The game begins with N coins. They alternate turns with Tony going first. Every turn, they must remove one, three or four coins from the table. The player who takes the last coin wins the game. For which values of N between 31 and 35 inclusive does Janet have a guaranteed winning strategy?

Update: Solutions are posted!