## Solution for #1

Johannes won; Rene came in second; Louis came in third.

## Solution for #2

The only lockers that remain open are perfect squares (1, 4, 9, 16, etc) because they are the only numbers divisible by an odd number of whole numbers; every factor other than the number's square root is paired up with another. Thus, these lockers will be "changed" an odd number of times, which means they will be left open. All the other numbers are divisible by an even number of factors and will consequently end up closed.

So the number of open lockers is the number of perfect squares less than or equal to one thousand. These numbers are one squared, two squared, three squared, four squared, and so on, up to thirty one squared. (Thirty two squared is greater than one thousand, and therefore out of range.) So the answer is thirty one.

Use the first two cuts to cut an 'X' in the top of the cake. Now you have four pieces. Make the third cut horizontal, which will divide the four pieces into eight. Think of a two by two by two Rubik's cube. There's four pieces on the top tier and four more just underneath it.

## Solution for #3

Use the first two cuts to cut an 'X' in the top of the cake. Now you have four pieces. Make the third cut horizontal, which will divide the four pieces into eight. Think of a two by two by two Rubik's cube. There's four pieces on the top tier and four more just underneath it.

## Solution for #4

## Solution for #5

Solution for #6

Not at all. He'll earn $5,368,709.12 on the thirtieth day alone.

## Solution for #7

Let *d* be the distance to the store, *T* be the time it gets to get there, *t* be the time it takes to get back, and *A* be the average speed (which is what we want to find out). As we know from elementary mathematics, distance equals rate times time:

d = 20T

T = d/20

d = 30t

t = d/30

Now that we have expressions for *T* and *t*, we can come up with an equation that describes the round trip:

2d = A(T + t)

2d = A(d/20 + d/30)

2d = A(3d/60 + 2d/60)

2d = A(5d/60)

A = 120d/5d

A = 24

So the average speed is 24 mph. If this seems strange to you, consider that more time is spent traveling at 20 mph than time spent at 30 mph, so the "20 mph" figure should count more toward the average.

## Solution for #8

Let *d* be the distance to the store, *T* be the time it gets to get there, *t* be the time it takes to get back, and *R* be the speed you travel on the return trip (which is what we want to find out). As we know from elementary mathematics, distance equals rate times time:

d = 20T

T = d/20

d = Rt

t = d/R

Now that we have expressions for *T* and *t*, we can come up with an equation that describes the round trip:

2d = 40(T + t)

2d = 40(d/20 + d/R)

2d = 40d(1/20 + 1/R)

1 = 20(R/20R + 20/20R)

20R = 20(R+20)

R = R + 20

Here we have worked our way into a paradox. The reason, simply, is that you have to travel back at an infinite speed to make your average speed 40 mph. This may seem strange, but consider that, the faster your return trip, the quicker you make it, and consequently, this faster speed has a lesser impact on the average speed.

If you traveled the return trip instantaneously, this would be equivalent to traveling double the distance in the same amount of time as the one-way trip. So if the rate of speed of the return trip is infinite, you do indeed get an average speed of 40 mph.

## Solution for #9

There is only one solution, discounting mirror image solutions and rotations:

Solution for #10

Since the trains are 100 miles apart, and the trains are traveling toward each other at 40 and 60 mph, the trains will collide in one hour. The bird will have been flying for an hour at 90 miles per hour at that point, so the bird will have traveled 90 miles.

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