Earlier today I set you these puzzles for English majors, i.e people who studied English at university. Here they are again with solutions and commentary from Ben Orlin, whose book Math for English Majors is out in September.
For each question below, which option is bigger? No calculators allowed!
1. Square vs cube
the sum of all squares from 1 to 100
the sum of all cubes from 100 to 200
[Note: squares are the numbers 12, 22, 32, … and cubes are 13, 23, 33, …]
[Update: There was a typo in the setting of this question. Sorry everyone. Now it is fixed]
Solution the squares
It’s tempting to pick the latter. Cubing grows a number faster than squaring it. Plus, the individual numbers in that range are bigger. Alas, that’s precisely the problem. Because squares grow slowly, the first sum includes 10 squares. But because cubes grow so quickly, the second includes just one cube: 53 = 125. Thus, the first sum is much larger.
2. Sir Pentages
17% of 32
32% of 17
Solution They’re the same
Each is 17 x 32 / 100.
3. Fraction infraction
3997/4001
4996/5001
Solution 3997/4001
Despite its many merits, the language of fractions has a fundamental problem: It’s hard to tell which of two is bigger! In this case, each of the fractions is a tiny bit smaller than 1. The first is 4/4001 shy, and the second is 5/5001 shy. Set aside the original fractions, and focus on these “missing piece” fractions. Rather than giving them a common denominator, go for a common numerator: the first is 20/20,005, while the second is 20/20,004. Each offers 20 pieces from a pizza cut into exceedingly thin slices. The first pizza, with its one extra cut, has slightly thinner slices, and thus, the first missing-piece fraction (4/4001) is just a bit smaller than the other. Thus, the first of the original fractions (3997/4001) is just a bit bigger. Indeed, if you convert them to decimals, you’ll see the fractions are identical for the first seven places, differing finally at the seventh: 0.99900025 vs. 0.99900020.
4. Roots shmoots
the square root of 6
the cube root of 15
Solution cube root of 15
These numbers are close together (differing by less than 0.02). They are also, to use a crude epithet for irrational, nasty. We need an operation that can simultaneously magnify their difference, and clean them up into a nicer form, without altering their relative size. The first number is begging to be squared. The second is begging to be cubed. So why not both? Raise the numbers to the sixth power! The first number becomes 63, which is 216. The second becomes 152, which is 225. Thus, the second number is larger.
5. Tick tock
the number of seconds in a year,
the number of hours in a millennium
Solution the number of seconds in a year
Instead of cranking through the tiresome multiplication (60 x 60 x 24 x 365 vs. 24 x 365 x 1000), let’s just focus on how the numbers differ. The second time span is 1000 times longer. The first ticks 60 x 60 = 3600 times more frequently. Thus, the first number is 3.6 times larger.
6. Power shower
2100
545
Solution 545
It’s annoying to compare a long string of 2 x 2 x 2’s with a long string of 5 x 5 x 5’s. So let’s convert those 5’s, as best we can, into 2’s, by rewriting each one as 2 x 2 x 1.25. Rearranging those factors, the second number becomes 290 x (1.25)45. Since the first number can be rewritten as 290 x 210, we’ve now simplified our comparison to this:
Which is larger, 210 or (1.25)45?
I.e., which is better: doubling ten times, or growing by 25% forty-five times?
If you have any experience with compounding growth, you’ll suspect the latter is better. Indeed, it is. Four consecutive increases of 25% is better than a single doubling. Now, we’re comparing more than 11 of these quartets to just 10 doublings. Thus, 545 is larger! By a factor, it turns out, of roughly 22.
Thanks to Ben Orlin for suggesting today’s puzzles. Math for English Majors is out on September 26.
I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
My new book, Think Twice: Solve the simple puzzles (almost) everyone gets wrong (Square Peg, £12.99), is out on September 5. To support the Guardian and Observer, order your copy at guardianbookshop.com. Delivery charges may apply.
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