 ponder this: The next sentence is true but you must not believe it. The previous sentence was false. Keep pondering.
 are there more even numbers or integers?
 is .9 repeating equal to 1? (hint: is .3 repeating equal to 1/3? triple those numbers.)
 what's wrong with this proof (other than the final outcome)?
a = b  a^{2} = ab  a^{2}b^{2} = abb^{2}  (a+b)(ab) = b(ab)  (a+b) = b  a+a = a  2a = a  2 = 1 
 are there more rational or irrational numbers?
 what would a sphere passing through 2D space appear as to a 2D creature?
 what would a 4D figure passing through 3D space appear as to a 3D creature?
 would an alien race, unfamiliar with our number system be able to understand the significance of this sequence, perhaps represented as a series of groups of objects of increasing quantity: 2,3,5,7,11,13,17,19,23,29... Do you understand this sequence?
 what symbol comes next: O,T,T,F,F,S,S,E,N,T...
 extend the Fibonacci sequence out to 20 terms, and divide the last term by the penultimate term. What ratio do you get, and why? Now try this same task with any two starting numbers. Really.
 what does this equal: 4  4/3 + 4/5  4/7 + 4/9  . . . (but why?)
 why (mathematically) are rectangular buildings 'easier' to build than pentagonal, hexagonal, etc. What is so fundamental about the perpendicular?
 a tube of 3 tennis balls...which dimension is larger, its height or circumference?
 what is the pattern of repeating digits in the decimal representation of 1/7? 4/7? (but why?) Try this with a denominator of 13.
 can you graph this: f(x) = (2)^{x} (why or why not?)
 look at these videos: http://www.youtube.com/user/Vihart
 why did the Babylonians use a base60 number system?
 what IS a base60 number system?
 what would a base1 number system look like?
 can you have a basen number system, where n is a fraction? 0? negative?
 ponder this conundrum: Three guests check into a hotel room. The clerk says the bill is $30, so each guest pays $10. Later the clerk realizes the bill should only be $25. To rectify this, he gives the bellhop $5 to return to the guests. On the way to the room, the bellhop realizes that he cannot divide the money equally. As the guests didn't know the total of the revised bill, the bellhop decides to just give each guest $1 and keep $2 for himself. Now that each of the guests has been given $1 back, each has paid $9, bringing the total paid to $27. The bellhop has $2. If the guests originally handed over $30, what happened to the remaining $1?
 when you're multiplying by hand 2 twodigit numbers, why do you shift the second product over a space?
 try some long division using Roman Numerals
 assume there are a finite number of uninteresting numbers; therefore there must be a lowest number in that set, which gives that particular number a very interesting quality, meaning it can't be in the set of uninteresting numbers. Must all numbers therefore be interesting? If so, what is interesting about, say, 57,823,897?
 consider multiplication as being a shortcut for repeated addition, and exponentiation being a shortcut for repeated multiplication. Is there a shortcut for repeated exponentiation? Addition is the shortcut for repeated....what? Is there a shortcut for repeated subtraction or division?
 what is the least number of colors required to shade in a map, so that no two adjacent regions have the same color? Try coloring in the 50 United States or the 64 Colorado counties with as few colors as possible.
 how large of a group of people do you need to have to have a better than even chance of at least 2 people sharing a birthday?
 if you have a group of 183 people (half the number of days in a year), what is the probability that at least two of them share a birthday?
 what are the 5 regular polyhedra? why can't there be any more than this?
 you've probably thought about 2D or 3D or 4D space. What would 2.87D space 'look' like?
 is math an art or a science? or something else?
 you've heard of square numbers. what are triangular numbers? pentagonal numbers? is there some sort of Pythagorean Theorem that would work with triangular numbers instead of square numbers? Would expanding the Pythagorean Theorem into 3D space make the exponents 3 instead of 2?
 one informal way of describing prime numbers is as the set of numbers that refer to quantities of objects (pebbles, say) that can't be arranged into a rectangle of height greater than 1. But what if we move beyond quadrilaterals to pentagons, triangles, hexagons, etc. Are there different 'types' of prime numbers that can't be arranged into these shapes?
 12+34+5+67+8+9=100. Find another such representation of 100 with 9 digits in the proper order.
 easy: among all the possible closed shapes you could draw that have equal area, which has the least perimeter? Hard: is there a possible shape with the largest perimeter?
 divide the entire surface of a sphere into two triangles. Really.
 would you rather have a clock that loses 1 minute a day, or doesn't move at all?
 the number 1 has its own wikipedia page. So does the number 2. What's the smallest integer that doesn't have a wikipedia page? Go make a wikipedia page for that number. :)
 two 100 foot tall towers have a 190 foot long cable strung between them, that sags down to a point 5 feet above the ground. How far apart are the towers?
 how many people must you have together to ensure that there is a subset of 3 people who all are mutual acquaintances, or mutual strangers?
 can you draw a triangle with more than 180 degrees in its angles? How high can you get? Consider a triangle on the surface of a sphere for this one. How about a triangle with fewer than 180 degrees in its angles? What kind of surface would you need to be on for this?
 draw a triangle on a piece of paper. Rip off the corners and assemble them 'together'. What do you get? What does this tell you?
 try to make a triangle out of these three lengths: 3, 5, 9. Why can't this be done? Can you develop a rule? How about an analogous rule for quadrilaterals? pentagons? ngons?
 a quadrilateral has 4 sides, a triangle has 3 sides; how about a bilateral (biangle?)? In spherical geometry, does a circle have one side? or infinite sides?
 why isn't a triangle called a trilateral?
 explain why raising a positive number to the 1/n is equivalent to taking the nth root of that number. Does that equivalence make fundamental sense, or did mathematicians just decide it to be so? Go through a similar process with negative exponents.
 what is 0^{0}?
 you have two envelopes, and inside each I have put some money. In fact, one envelope contains twice as much money as the other. I'll let you select one envelope, which you can have after the game is over. But as soon as you select one, I offer you the option to switch envelopes. Should you switch? Why?
 consider a space with a finite area yet an infinite perimeter. Is this possible? (yup) Does that mean that you could paint the interior with a can of paint, yet it would take you an infinite number of pencils to trace the edge of this shape? (yup)
 an oldie but goodie: "I'm lying right now."
 google/explore the Monty Hall paradox.
 pick a number between 0 and 1 at random. What is the probability that number is rational? (0%) What is the probability that number is irrational? (100%). How can this be so? (if you think that you picked a number at random and you picked a fraction, you didn't really use randomness). How could you pick a totally random number?
 Draw a number line, labeling the points 1 and 10, with no tick marks or labels between. Point to a location evenly spaced between 1 and 9. What is it? (5?) Now do this same experiment with a young child. What point do they pick? (perhaps 3?) Why? (hint: do you think arithmetically or logarithmically? young children think logarithmically.)
 Collect a bunch of natural data. A whole lot of natural data. Perhaps the size in acres of every watershed in the world, or the height in centimeters at a moment in time of every oak tree in the world. More of the data points will start with 1 than 2, more will start with 2 than 3, and so on, with the fewest number of data points starting with 9. How can this be so? (explore Benford's Law). Does this mean that there are more 1's than 9's out in the universe? Dang.
 e^{i*pi}=1 (but why?)
 One day, you walk up a mountain. You set off at 9AM, and reach the top at 4PM. You take your time, sometimes stopping, sometimes going faster, sometimes going slower. You have a nice walk. You spend the night on the top of the mountain, enjoying the good views. The next day, you start back down, starting at 9AM and reaching the bottom at 4PM. Again, you take your time, stopping to rest periodically, going faster at times, going slower at times. You speed up, slow down, stop at different places than on your way up. Your pace is pretty random. But on both the ascent and the descent you always stay on the trail. Question: is there a point on the trail that you reach at exactly the same time on both days? Sometimes? Always? Never? Think think think!
 what comes next: 1, 5, 32, 288, 3413, 50069, ...
 You are a prisoner sentenced to death. The Emperor offers you a chance to live by playing a simple game. He gives you 50 black marbles, 50 white marbles and 2 empty bowls. He then says, "Divide these 100 marbles into these 2 bowls. You can divide them any way you like as long as you use all the marbles. Then I will blindfold you and mix the bowls around. You then can choose one bowl and remove ONE marble. If the marble is WHITE you will live, but if the marble is BLACK... you will die." How do you divide the marbles up so that you have the greatest probability of choosing a WHITE marble?
 Check out this website: http://htwins.net/scale2/
 Suppose you had a rope stretching around the entire world, laying flat on the ground. It would be 40,075 km long. Suppose you suspended the rope a meter off the ground at all points. How much would you have to add to the rope?
 Enjoy this animation (and what would a comparable animation for tangent look like?)
Unsolved Problems (to me, at least)  you have 5 identical dimes, 6 identical quarters, and 10 identical pennies. How many different groups of 5 coins can you make? Swapping out a dime for another dime wouldn't change the group, and the order in which they are selected doesn't matter. I'm most interested in an algorithm that would solve a more complex version of this problem  just an organized list of possibilities isn't an ideal solution. This is analogous to the distinguishable permutation problem, for which there is a simple formula; here, however, order doesn't matter: this is a distinguishable combinations problem.
 You've entered a lottery in which there are 35 winners chosen. There are 619 total entrants, with varying numbers of tickets from 1 up to 64. There are 1801 total tickets. You have 2 tickets; what is the probability that you get selected? (I think that this depends on the specific arrangement of tickets distributed  ask me if you want those data, or consider just starting with a smaller analogous problem; this is the way in which entrants were chosen for the 2013 Hardrock 100 ultramarathon).
