Thursday, April 2, 2015

Googoling the Answer

Here's this week's NPR Sunday Puzzle:
[Warning:] it's a little tricky. Given a standard calculator with room for 10 digits, what is the largest whole number you can register on it?
At least there is only one right answer. What we've got--and what we think Will Shortz believes is the "right" answer--is this:

which is 9999 GOOGOL (or 7060066666 upside down). GOOGOL is ten to the power of one hundred. Wiki explains all about it. If there was a way to get GOOGOLPLEX on the calculator, that would be even better.

9999 GOOGOL is not the largest number, though. Henry BW has thought of a larger one but we suspect Will won't accept Henry's answer, even assuming someone (Henry?) sends it in. That answer is this (I asked for a pretty picture but I may not get one in time):
On scientific calculators, the symbol E is standard for “times 10 to the power of”.
If you key in 6636363636 and then turn the calculator upside-down, you get:
9E9E9E9E99, which expands to:
9 x 10^(9 x 10^(9 x 10^(9 x 10^99)))
which is a lot.

Though if you insist on reading it from the other end as:
(((9 x 10^9) x 10^9) x 10^9) x 10^99
Then I think that
(which is 7090096636 upside down) and which expands to:
9 x 10 ^ (99 x 10^ 100)
is larger. 
Good to know. What did you all get?

I asked Flickr for pictures of GOOGOL (or GOOGOLPLEX; I wasn't fussy). Here's what I got:

Time for

Here are this week's picks:
Zero and fewer
    1 - 50 -- Word Woman
 51 - 100 -- Curtis
101 - 150 -- Natasha
151 - 200 -- legolambda
201 - 250 -- B. Haven
251 - 300 -- Joe Kupe
301 - 350 -- Marie
351 - 400 -- Maggie Strasser
401 - 450 -- Ross
451 - 500 -- Magdalen

501 - 550
551 - 600
601 - 650 -- Mendo Jim
651 - 700
701 - 750
751 - 800
801 - 850
851 - 900
901 - 950
951 - 1,000
1,001 - 1,050 -- David
1,051 - 1,100 -- Henry BW
1,101 - 1,150
1,151 - 1,200
1,201 - 1,250
1,251 - 1,300
1,301 - 1,350
1,351 - 1,400
1,401 - 1,450
1,451 - 1,500

1,501 - 1,550
1,551 - 1,600
1,601 - 1,650 -- Margaret G.
1,651 - 1,700
1,701 - 1,750
1,751 - 1,800
1,801 - 1,850
1,851 - 1,900
1,901 - 1,950
1,951 - 2,000
2,001 - 2,050
2,051 - 2,100
2,101 - 2,150
2,151 - 2,200
2,201 - 2,250
2,251 - 2,300
2,301 - 2,350
2,351 - 2,400
2,401 - 2,450
2,451 - 2,500

2,501 - 2,750
2,751 - 3,000
3,001 - 3,250
3,251 - 3,500
3,501 - 4,000
4,001 - 4,500
4,501 - 5,000

 > 5,000
 > 5,000 + new record
Our tie-break rule:   In the event that a single round number is announced with a qualifier such as "about" or "around" (e.g., "We received around 1,200 entries."), AND two separate people picked the ranges of numbers just before and just after that round number, the prize will be awarded to whichever entrant had not already won a prize, or in the event that both entrants had won a prize already or neither had, then to the earlier of the two entries on the famous judicial principle of "First Come First Serve," (or in technical legal jargon, "You Snooze, You Lose").  As of July 2012, this rule is officially no longer obsolete (and also I still just like having fine print).


Anonymous said...

Do I sense dissatisfaction with the answer? I don't know what would dismay me more--your being right about 9999googol (which had occurred to me) or my paranoid suspicion that the answer is 8. Both are just wrong.*


* One clearly and the other based on my neither disinterested nor uninformed opinion.

Alex B. said...

There's no way he accepts "8" because there's no possible argument that infinity is a whole number. In fact, you could argue that a googol isn't a whole number either, if you want to define a whole number as "a number that can be used for counting", since there isn't a googol of anything. (The number of atoms in the universe is somewhere around 10^80.)

Still, I like the answer of just "googol", I think it's clever.

Paul said...

I think a standard calculator is one of the type used in a regular classroom. I haven't been in a regular classroom in decades, nor do I have ready access to a 10-digit calculator. Just for giggles, I rounded up an old cell phone with an 8-digit calculator on it. I multiplied 99999999 by itself until I got to 9.9999988e+95 and multiplied by 99999999 again to get Error. Maybe that's Will's intended answer. After all, there are some pretty goofy names for large numbers.

Paul said...

Hmmm ... what's the average number of quarks per atom?

Dave Taylor said...

The plural of "googol" is "googols",

so I think the largest theoretical 10 digit representation might be:

9e9 googols (5106006636)
or approx 3.9 x 10e108

However, no-one says 10 ThousandS we just say 10 Thousand when counting objects, so now I'm not sure...

Paul said...

And how many square Planck lengths of space does a quark occupy?

Paul said...

Cubic ... I meant cubic ... I knew that.

Paul said...

Anonymous said...


I agree that infinity is not a whole number. (Oddly, though, it now appears to be a real number according to the College Board, which administers the SAT.) But of course a googol is a whole number. Whether it applies to the real world is irrelevant, at least if one takes the whole numbers to be the nonnegative integers, which is the standard definition.

Then again, this puzzle is one of ambiguities, and even the first image (of the 1970s Sierra Club ad) is wrong, as has been pointed out here.

Oh well. At least I don't have to explain why "googol" is correct and "9999googol" is wrong or the reverse. Then again, neither does WS.


Anonymous said...

What did I come up with?
I came up with a vision of Mrs. Shortz yelling "Willy, if you don't turn that damn calculator right side up and do your homework, you are going to be in sooo much trouble."

I think I am going to change the question I will ask when I am on the show from "How much does NPR pay you for this gig?" to "What will it take to convince you that your millions of listeners are sick and tired of upside down digital displays?"

Getting this puzzle mainly involved giving up on googolplex.

I am not convinced that a number/letter combo works.
Mendo Jim

Dave Taylor said...

I'm pretty sure TPM is thinking 9999googol, as it seems the "most elegant" solution requiring a 10 digit calculator.

However, the most elegant answer may not always be the most accurate...

Natasha said...

I was searching for a 10 digit answer based on the working of the puzzle. I believe there is a 10 digit answer as stated in the comments above.

Word Woman said...

Natasha, my bet is on 9999googol as well.

Seth said...

I think 9999googol is wrong. There should be a space between 9999 and googol, which can't be expressed on a calculator. You don't say 99thousand, for example.

I like 9E9E9E9E99. I think Will is thinking of googol, but this construction with the E's should count and win.

Henry BW said...

But infinity is problematic, because there are different infinities. I tried and failed to do aleph-0, which is the smallest countable infinite number.

legolambda said...

On Sunday's thread I commented:
"I have one really tricky answer which may actually be Will's/Henry Hook's intended answer, or may just be another defective by-product of my delusional mind! Still, this is the one I am submitting."

I did submit it. I did not "get the call" from Will (actually from a youthful NPR assistant).

Also in Sunday's thread, "anonymous" (Phil? Mendo Jim?) predicted that finally we would hav a puzzle with an answer the same as its range. If my answer is Will's intended, that prediction may well come true.

I will reveal, explain and discuss my answer on tomorrow's edition of Puzzleria! (see this site's "Blogroll") which I plan to upload in about 12 hours of fewer.


Natasha said...

correction: wording not working (in comment above)

Anonymous said...

I'm reminded of a previous time when there was a lot of complaining about a solution before it was revealed. (If I recall correctly, it was with a film title.) Perhaps WS's answer has nothing to do with googols, infinities, or anything. Maybe he's got something simpler like 6666666636, which becomes
9Ɛ99999999 when written (or viewed) upside-down. Just a thought.


(˙ɯoɔ˙ʇxǝʇuʍopǝpᴉsdn ɟo ʎsǝʇɹnoɔ ƃuᴉpɹoʍ uʍop ǝpᴉsd∩)

legolambda said...

Phil, Thanks for the nifty inverted-text tool! I will use it to make a pineapple cake! or create some upside-down digital clock puzzles... can never have enough of those!

Speaking of puzzles, every year around this time Will Shortz springs one of his "two-week creative challenges" on us unsuspecting inverted-digit-Roman-numeral-loving-anagram-crazy-movie-actor-fanatic puzzle solvers.

Well, this year I have beat him to the punch (Sangria spiked with vodka!) with a creative challenge of my own on Puzzleria! It involves a delightful activity I call "acrodefining," a hybrid of acronyms and definitions.

For example, "PHIL" could be acrodefined as "Perfectly Honest Ingenious Linguist" or "Praiseworthy Honorable Inversion Lover."


Anonymous said...


Thank goodness you didn't opine "plays hockey in lingerie." I don't want all my secrets revealed.


Anonymous said...


I read your explanation for 99, but why not make your fingers binary digits and count up to 1023?

On a more serious note, if "calculator" does mean the hands, then isn't the puzzle seriously flawed since the clever solution (i.e., using the hands) involves a smaller number?


legolambda said...


I wasn't sure if it was hockey or Equestrian Rugby that you played in lingerie, so lest be haunted by the PERIL of being wrong, I didn't go there.

Thanks for visiting Puzzleria!

Regarding your more serious note, you have stated this puzzle's dilemma perfectly. Indeed 10, 99 (or even 1,023) pale in comparison to the googols of this week's puzzle world. And that is exactly why I am so curious about this morning's reveal.

That's because I have a real hunch that "calculator = hands, etc." is Hook's tricky answer. But all our googol answers satisfy the puzzle's wording and, of course, blow "hands" out of the water size-wise.

And those of us who submitted "googol solutions" could justifiably argue that the old "inverted digit trick" is "a little tricky" while the somewhat more novel "hands = standard calculator" is "a bit more than a little tricky!"

Which would put Will in a bind. If my hunch about Hook's intended answer is correct, I believe will will accept 10, 99 or whatever Hook says is correct regarding "hands." But unless he want a mutiny on his hands, he will also accept what he deems the largest legitimate entry among all our gargantuan googol and E-exponent guesses.

In short, I think Will will accept 99 and 9999googol as the correct answers.

I loved this puzzle.


legolambda said...

I just heard the answer to last week's puzzle. I was wrong about the "hands = calculator" answer. I had a hunch. I was dead wrong, just as I was wrong about last autumn's "four clock-times" puzzle which I thought was "FALL BACK" was was really "the hands resemble the Roman numerals C, I, V and L."