HOW TO BE A GOOGOLAIRE
(CONSTRUCT A GOOGOL WORLD IN A SHOEBOX)

When our first child, Tim, was two, I told him the story (given in an associated file) of how a five-year-old boy made up and named the "googol": one followed by one hundred zeros.

Tim asked it "there were a googol of people". "No."

After going to the beach in Puerto Rico, where we lived, he asked about "a googol of grains of sand". "No."

The Puerto Rican sky, not polluted by ground-light glare of The Continent, shows many more stars than we see in The States. So Tim asked about "a googol of stars". "No -- not in the enterire universe."

After Tim had heard about electrons, Tim asked about "a googol of electrons". "No -- only about half that many electrons -- in the whole universe. There isn't a googol of ANYTHING. It's just a big number -- thought up by a little boy."

Two years after these googol-discussions began, his brother, Chris, was born when I was involved in some of the math that later involved "The New Math" -- in particular "Boole's Theorem", described below. I realized that THERE ISN'T A GOOGOL OF ANY THING -- NOTHING MATERIAL. But IT'S EASY TO SET UP A SITUATION INVOLVING A GOOGOL OF CHOICES.

Any child, however poor, can BECOME A GOOGOLAIRE, incredibly more than A MILLIONAIRE!

Hence, this model, which I later demonstrated to Third-Graders at our Campus School.

MODEL

  1. On a sheet of lined tablet paper (one of many needed), draw the vertical lines composing 9 columns.

  2. Label the columns, consecutively, T1, T2, T3, T4, T5, T6, T7, 78, T9, to represent 9 different types OF ITEMS.

  3. You are going to use many pages of lined tablet paper so that each row represents a CHOICE, either involving a particular TYPE or NOT.

  4. The intersection of a column (for TYPE) with a row (for CHOICE) isolates a CELL. You are going to write either "1" or "0" in each cell of this and other needed tablet papers. Why "1" or "0"? Well suppose the 15th row has a "1" in column, say, "T4": that means that the "15th choice involves an item with atribute of type T4". If the 15th row has a "0" in, say, column, "T7", this means that the item that is the 15th choice DOES NOT HAVE THE ATTRIBUTE labeled "T7". Dig?

  5. You now fill in 512 rows with a ONE or ZERO to a column, on this page and on other lined tablet pages with the same column headings. (Why 512? Because 512 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 -- product of nine twos. Because each type, T1, T2, etc., is EITHER CHOSEN ("1") or NOT CHOSEN ("0"), resulting in 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 512 POSSIBILITIES. This is written, as "2 to superscript 9".) Not to worry! You can "mechanically" write out these rows by a COMBINATORIAL ALGORITHM.

  6. You now have a given number of lined tablet papers with the column headings (TYPES) and 512 rows (CHOICES). We now want to get at the CHOICES themselves. To do so, we SCISSOR OUT THE 512 ROWS (AS CHOICES).

  7. We put these 512 slips into a shoebox to DRAW FROM.

  8. HOW MANY DRAWS HAVE WE? We can either draw 0 slips from the box; or 1 slip; or 2 slips; or -- etsettery -- up to all 512 slips. OUR DRAWINGS ARE INDEPENDENT! That is, if -- at one time -- I draw 4 slips -- then, later, 9 slips -- these two drawing are INDEPENDENT -- SINCE I PUT BACK THE 4 SLIPS FROM THE FIRST DRAWING -- AND THE BOX NOW "HAS NO MEMORY" OF THAT DRAWING -- HENCE, THE NEXT DRAWING IS INDEPENDENT OF THE PAST! That tells us how many DRAWINGS. For each of the 512 slips, there are exactly 1 possibilities: DRAWN or NOT-DRAWN. So WRITE 2 DOWN 512 TIMES AND MULTIPLY IT OUT, FOR ALL THE POSSIBILITIES OF NO SLIP UP TO 512 SLIPS IN A DRAWING. This number can be written as "2 superscript 512", which I'll show to be vastly more than a GOOGOL.

  9. How do you convert "2 to superscript 512" to a decimal form of "10 to superscript x", for some number "x"? There's a trick from logarithms that shows that "x", APPROXIMATELY, to be "3-tenths of 512". Now, (.3) x 512 = 153.6, a number between 153 and 154. So. decimally. THE NUMBER IS BETWEEN "10 SUPERSCRIPT 153" AND "10 SUPERSCRIPT 154" -- that is, THE NUMBER OF CHOICES IS BETWEEN ONE FOLLOWED BY 153 ZEROS AND ONE FOLLOWED BY 154 ZEROS -- VASTLY MORE THAN A GOOGOL!
EXAMPLE OF DRAWING