1) Is 1050 half a googol? Why not?
2) The number of atoms in the known universe is something on the order of 1080, while the number of stars in the universe is estimated to be around 5 X 1022. What happens when you add those two quantities together? Shouldn't it be enough to make more than a googol?
3) OK, I'll spot you an extra universe. Double that quantity above and see if gets you to a googol.
4) Perhaps I was not generous enough. OK, take a million universes. Oh, what the heck-- use a billion. See if that gets you to a googol.
5) Exponents are weird!
6) How many different ways can the members of your group sit around the table?
7) Moving people around the table is essentially like shuffling cards. How many different ways could you shuffle a deck of fifty-two cards?
8) My sister and I frequently play a double solitaire game. The amazing thing is that we never get tired of it, even if we play it several times in a row. But then again, it's not so amazing, since each game is unique. The likelihood of us ever seeing the same game twice is extremely low.
In this particular version of solitaire, all of the cards are laid out at the beginning (no leftovers). How the game will play out depends on how the cards are shuffled. A particular shuffling of the cards will lead to my cards being laid out in a particular arrangement. Changing even a single card could change the whole game for me. The complexity of the game comes not just from how my particular arrangement of cards works out, but also from how my arrangement is paired with hers. So I might be playing with shuffling #10113, while my sister has shuffling #222504431. That's a game.
How many different games are possible?
9) At a friend's house recently, we played a game of "Hearts." Since we had nine players, we decided to shuffle together two different decks of cards. Once the cards are shuffled, they are stacked in some random arrangement. We decided to make each deck's cards distinct. One deck had boats on the back and the other had horses. We made it so that the boat queen of spades would be higher than the horse queen of spades. How many such arrangements are possible? Is this the same problem as #6?
10) We played some other games with two shuffled decks. In another game, it didn't matter which deck your card came from; getting the horse queen of spades was the same as getting the boat queen of spades. What does this do to the number of possible arrangements of the two shuffled decks?
11) Some computations are difficult to do, especially when the numbers get so large that they exceed the capacity of an ordinary calculator. Go ahead-- put in 104 factorial. The calculator will probably say "E" or "overflow". (Even if yours doesn't do that, pretend it did.) You will have to be more clever than the machine. Find a way to compare 104! to a googol. Can you say for certain which quantity is greater, just by thinking about the way each number is built?
12) Sometimes one of my students will express a desire to be a googolaire-that is, to have a googol dollars. In fact, this would be an enormous burden. If you kept it in $10,000 bills, how much space would you need to store a googol dollars?
13) Are you impressed with how big a googol is? Notice how it vastly exceeds anything we tried to count, like heartbeats or grains of sand. The sand group's computation came out to something on the order of 10^26. Is that one-fourth of a googol? No! What if you had a trillion earth's worth of sand? Big deal! We're still only up to 10^38. And yet (dun dun dun)…
You can hold a googol in the palm of your hand. Just shuffle together two decks of cards and think about how many possible shufflings there are. Think about the probability that your shuffling matches my shuffling. Less than 1 in a googol. Not very likely!
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