Puzzle Answers

1. Twelve thousand and twelve dollars, or eleven thousand eleven hundred and eleven dollars?

The first number is 12,000 + 12. The second number sounds smaller until your ear catches that sneaky phrase “eleven hundred”; it turns out that 11,000 + 1100 + 11 (which equals 12,111) is the larger of the two.

2. 19/200 of this pie, or 29/300 of it?

Each fraction is slightly smaller than 1/10 of the pie. The first is 1/200 shy; the second is 1/300 shy. Thus, the second fraction is closer to 1/10, and is therefore larger.

3. One kilogram of quarters, or twenty-five kilograms of pennies?

If every kilogram contained, say, 1,000 coins, then the two bundles would be equal in value. But pennies are far lighter than quarters, so you get more of them per kilogram. Thus, 25 kilograms of pennies are more valuable (albeit less convenient for shopping).

4. A penny for every second in a month, or a penny for every hour in a century?

The question can be paraphrased: To maximize ticks, would you rather your clock tick more often (every second vs. every hour) or for a longer time period (a century vs. a month)?

Since there are 60×60 seconds in an hour, the first clock ticks 3,600 times as often as the second clock.

Since there are 12×100 months in a century, the second clock ticks 1,200 times longer than the first clock.

The first value, then, is larger by a factor of roughly 3 (that is, 3600/1200), the precise value depending on which month you pick.

5. The tenth root of 10, or the cube root of 2?

These numbers are quite close together. To tell them apart, we need a way to exaggerate their differences. My proposal: Let’s raise each one to the 30th power! (This is the cube of the tenth power.)

The first number becomes the cube of 10, thus 1,000.

The second number becomes the tenth power of 2, thus 1,024.

So the second number is slightly larger.

6. $10 plus half of the second, or $20 minus half of the first?

Let’s call the value of the first envelope x.

The second envelope has 20 – 0.5x.

Meanwhile, x is equal to 10 + half of this amount.

Thus,              x = 10 + 0.5(20 – 0.5x).

This means    x = 10 + 10 – 0.25x.

Thus,               1.25x = 20.

This gives us  x = 16, and so the second envelope has $12. The first is worth more.

7. In our first year, we gained 90 percent. In our second year, we lost 50 percent. Would you rather have the amount we originally invested, or our current value?

Doubling and then losing half puts you right back where you started. But this fund didn’t quite double; instead of gaining 100 percent, it gained only 90 percent. So when it lost 50 percent, it wound up below the originally invested value. (Specifically, 5 percent below.)

Bonus: Why does the Lesser Fool always take the smaller amount? And who might be the “Greater” Fool?

The Fool takes the smaller amount because that’s how the con works! If he started taking the larger one, he’d cease to be remarkable, and that would bring an end to the parade of people offering free money. These people, naturally, are the Greater Fools.

— Ben Orlin

1. What year would it be right now? For that matter, what century would it be? 

It is the year 2025, which means 2 thousands, 0 hundreds, 2 tens, and 5 ones. In the zero-less system, we cannot have zero hundreds; one of those thousands must be unboxed as ten hundreds. Thus, the current year would be 1T25. 

And what century is it? It’s the 1TXXs, which you might call “the ten-teen hundreds.” (We can’t say 1T00s, of course, because that would use the forbidden symbol.) 

Note, with perplexity, that the present century began not 25 years ago in 19TT (formerly 2000), but just 14 years ago in 1T11 (formerly 2011). And it will continue beyond the year 1T9T (formerly 2100), ending only in the year 2111 (a year that, at last, requires no translation.) 

2. Would a “six-figure salary” be more or less desirable than under the old system? 

Under our system, six-figure salaries go from $100,000 to $999,999. 

Under the new system, they go from $111,111 (the smallest six-digit number) to $TTT,TTT (the equivalent of our $1,111,110). 

So, a six-figure salary is more desirable under the new system. 

3. Map out the ways a zero-less culture would differ. Would towns commemorate 111th anniversaries? On a car’s odometer, which mileage rollover would be most exciting? And would anyone care that Wilt Chamberlain once scored 9T points in a basketball game? 

Your guesses are as good as mine, but I would wager on the following. 

1. Just as we currently make a big deal out of birthdays and anniversaries ending in zero, we’d do the same for those ending in T. So would we celebrate a centennial on the 9Tth? No, my hunch is that the TTth (what we call 110th) or the 111th would be the big ones. 

2. On a car’s odometer, it’s pretty clear to me that the coolest rollover is when you reach 111,111 miles, going from TT,TTT to 111,111. 

3. People would still care that Chamberlain scored 9T in a single game, because no matter how you enumerate it, that’s impressive! 

And as for other ways a zero-less culture might differ: 

4. We’d say that impressive sums of money have “lots of digits” (not “lots of zeros”). 

5. The temperature that’s so cold all molecules slow to a stop would be known as “absolute freeze” or “the all-frozen” (not “absolute zero”). 

6. The person from whom an infection spreads would be known as “the origin patient” (not “patient zero”). 

7. “Zero-sum” games would be known as “win-lose” or “perfect tradeoff” games. 

8. Game shows wouldn’t give away million-dollar prizes, since $999,99T is not a very cool-looking number. Instead, they’d give away $TTT,TTT or $1,111,111 prizes. 

9. Instead of reading “0–0,” the scoreboard at the start of a game would have two blanks. 

10. Learning arithmetic in school would be even more daunting than it already is! 

— Ben Orlin