Bosco

In a row of five squares, each square is to be colored either red, yellow, or blue, so that no two consecutive squares have the same color. How many ways are there to color the five squares, if there must be at least three yellow squares?    

In order not to have two yellows next to each other,    

the yellow color must occupy squares 1, 3, and 5.   

That leaves squares 2 and 4 to fill, and two colors    

to do it with. There are only four ways.   

Blue Blue     Red Red     Blue Red     Red Blue    

So, the answer is four ways.    

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I have $3$ different mathematics textbooks, $2$ different psychology textbooks, and $2$ different chemistry textbooks. In how many ways can I place the $7$ textbooks on a bookshelf, in a row?    

7 • 6 • 5 • 4 • 3 • 2 • 1  =  5040    

Seven books total, so you have 7 choices for the first book.    

For each of those choices, you have 6 choices for the second book.   

There are 5 choices left for the third book in the row.    

et cetera until all of the books are shelved.    

The expression for the arrangements can be shown as 7! (called seven factorial).   

Another way to express the number of arrangements is ₇P₇ (called seven permute seven).    

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The number $100$ has four perfect square divisors, namely $1,$ $4,$ $25,$ and $100.$    
What is the smallest positive integer that has exactly $2$ perfect square divisors?    

That would be 4.  Its two perfect square divisors are 1 and 4. 

We ignore 2 as a divisor of 4 because it isn't a perfect square,    

just as the example given in the problem ignored 10 and 20, etc.    

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For a positive integer $n$, $\phi(n)$ denotes the number of positive integers less than or equal to $n$ that are relatively prime to $n$.   
What is $\phi(191)$?    

191 is a prime number.  Therefore, it is relatively prime to every integer less than itself.    

You don't count 1, so there are 189 integers (i.e., 2 through 190) relatively prime to 191.    

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The number of diagonals in a certain regular polygon is equal to $2$ times the number of sides. How many sides does this polygon have?    

The number of diagonals in a regular polygon with "n" sides is calculated using the formula: n(n-3) / 2.    

             from the formula                                d  = n • (n – 3) / 2    

             given                                                  d  =  2 • n    

                                                                      2n  =  n • (n – 3) / 2   

                                                                      2n  =  (n² – 3n) / 2     

                                                                      4n  =  n² – 3n    

                                                                        n²  =  7n    

                                                                         n  =  7    

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Find the number of bases $b \ge 2$ such that $100_b - 1_b$ is prime.    

The third position to the left is the base raised to the second power position.    

So this problem could be viewed as "For which bases "b" can (b² – 1) be prime?"    

The only base I could find for which this works is 2 ... i.e., 2² – 1 = 3.    

Therefore, the answer to the problem is:  there is only one.    

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Let $IJKLMN$ be a hexagon with side lengths $IJ = LM = 3,$ $JK = MN = 3,$ and $KL = NI = 3$. Also, all the interior angles of the hexagon are equal. Find the area of hexagon $IJKLMN$.    

The sides are equal and the angles are equal.    

That means the hexagon is a regular hexagon.    

                                                                                               3 sqrt(3)    

The area of a regular hexagon in terms of its sides is   A  =  ––––––––  x  a²  where "a" is a side.    

                                                                                                     2    

                                                                                       A  =  ( 3 x sqrt(3) x 3² ) / 2       

                                                                                       A  =  ( 27 sqrt(3) ) / 2    

Is that good enough or do you need a number.     

                                                                                       A  =  ( 27 x 1.732 ) / 2     

                                                                                       A  =  23.383    

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A terminal zero is a $0$ that appears at the end of a number. For example, the number $3,800$ has two terminal zeros.   
How many terminal zeroes does $40 \cdot 6 \cdot 75 \cdot 12$ have?    

          40 • 6 • 75 • 12 = 216,000        so, three terminal zeros    

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Find the $4000$th digit following the decimal point in the expansion of $\frac{1}{117}$.    

The 4000^th digit is 5.    

1 / 117 in decimal notation has a repeating phrase. 

1 / 117 = 0.008547     

Let's count 'em up:         0.008547(6^th digit)   008547(12^th digit)   008547(18^th digit)   008547(24^th digit)  • • •       

                                          008547(3990^th digit)   008547(3996^th digit)   0085(4000^th digit)      

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