A man has to take a wolf, a goat, and some cabbage across a river. His rowboat has enough room for the man plus either the wolf or the goat or the cabbage. If he takes the cabbage with him, the wolf will eat the goat. If he takes the wolf, the goat will eat the cabbage. Only when the man is present are the goat and the cabbage safe from their enemies. All the same, the man carries wolf, goat, and cabbage across the river. How? 

Let A =  1, B =  2, and so on, with each letter equal to its position in the alphabet. The lexivalue of a word is the sum of the values of its letters. For example, ROMANS has a lexivalue of 18 + 15 + 13 + 1 + 14 + 19 =  80.

Now, do the following:

• Pick a number.
• Convert it into its representation in Roman numerals.
• Find the lexivalue for that Roman numeral.

For example, if you choose 11, that becomes XI in Roman numerals, and XI has a lexivalue of 24 + 9 =  33.

Are there any numbers for which the lexivalue is equal to the original number?

Equations to solve in your head:

Is this a joke? Not if you can multiply the first equation by 6,751 and the second by 3,249 in your head, and not if you use a second, simpler method.

Wheels A, B, C, and D are connected with belts as shown. If wheel A starts to rotate clockwise as the arrow indicates, can all 4 wheels rotate? If so, which way does each wheel rotate?

Can all the wheels turn if all 4 belts are crossed? If 1 or 3 belts are crossed?

A plywood sheet is 45 by 45 inches. What is the approximate diameter of the log the sheet was made from?

The diameter d of a circle equals , where C is the circumference, but please do not make a mistake. The diameter of the log is not .

How do I love thee?  Let me graph the ways! 

Can you come up with one or more equations to graph a heart on the coordinate plane? The equations can be rectangular, polar, or parametric.

Bonus: Can you shift your heart so the graph or its interior includes the point (2, 14)?

How do I love thee?  Let me plot the ways! 

A heart is drawn on a coordinate plane by plotting the following points and connecting them:

• The coordinates of the points are:

( n , n ), ( n - 3, n + 3), ( n - 6, n ), ( n - 9, n + 3), ( n - 12, n ), ( n - 12, n - 3), ( n - 6, n - 9), and ( n , n -3).

• The coordinates of one point are (2, 14).
• All coordinates are positive integers.

What is the value of  n ?

A magic rectangle is an m× n array of the positive integers from 1 to m× n such that the numbers in each row have a constant sum and the numbers in each column have a constant sum (although the row sum need not equal the column sum). Shown below is a 3 × 5 magic rectangle with the integers 1-15.

 Two of three arrays at left can be filled with the integers 1-24 to form a magic rectangle. Which one can't, and why not? 

How do I love thee?  Let me build the ways! 

Make a heart using any of the shapes in the PDF file. You can change their size, but you cannot change their shape. And you can use a shape more than once. 

Can you make a heart with just three shapes? What about five? Six? Ten? How many different hearts can you make?

The Fibonacci sequence is shown below, with each term equal to the sum of the previous two terms. If you take the ratios of successive terms, you get 1, 2, , , , , and so on. But as you proceed through the sequence, these ratios get closer and closer to a fixed number, known as the Golden Ratio.

1, 1, 2, 3, 5, 8, 13, …  

Using the rule that defines the Fibonacci sequence, can you determine the value of the Golden Ratio?

Ask a friend to pick a number from 1 through 1,000. After asking him ten questions that can be answered yes or no, you tell him the number.

What kind of Questions?