Test your quantitative reasoning

This October, my colleague Rich and I went to the National Numeracy Network annual meeting in Northfield, MN (more on that here). During a session on teaching quantitative reasoning led by the brilliant Kate Folliette, folks shared their favorite brain teasers.

A subset of my favorites are below, and are also currently posted outside of my office (ETC 225). Think through these questions; check the links for more resources/background. (Answers at end of post)

(1) Explain how this arithmetic is valid.
1 + 1 = 2
3 + 4 = 7
5 + 6 = 11
7 + 8 = 3
9 + 10 = 7
11 + 11 = 10

(Hint: you most likely use this math multiple times a day…)

(2) Car crashes part 1: Dangerous designated drivers?

1/3 of all car crash fatalities are caused by drunk drivers.

Therefore 2/3 of all crash fatalities are caused by sober drivers.

Are sober drivers twice as dangerous?

(3) Car crashes part 2: Astrology linked to crash rates?

This data from 2010 was released by Allstate, linking people in car crashes and their astrological signs.


The data is accurate; the joke was that Allstate was going to use astrological signs as a basis for assigning insurance rates. (Because of public outcry, they later issued a statement of apology and retracted the joke.)

I will tell you that there is no significant relationship between astrological sign and accident rate, but will reiterate that the data is accurate. Explain how this chart makes sense.

(4) Rough surfaces

Which is rougher, the surface of the earth or the surface of a basketball?

(First try to answer this question based on your intuition. For information to help you answer based on quantities, see here.)

(5) Series and patterns (diagram)

The five shapes shown below are the first five in a series. What comes next? diagram1

————– (answers below) —————

(1) 1+1 = 2, 5 + 6 = 11, 11 + 11 = 10 …

Unless you are accustomed to rising at Oh-Six-Hundred, you use this counting system daily if not — ahem — hourly. One hour after 1pm is 2pm, five hours after 6pm is 11pm, eleven hours after 11am is 10pm.

This is modular arithmetic; in this case, “Mod 12”. If 11 + 6 = 3, you’d be working in Mod14; the modulus is where you start over in your numbering system.

(2) 2/3 of car crash fatalities are caused by sober drivers…

Are sober drivers twice as dangerous? Maybe. We need more information to answer this question.

Of the entire group of people who drive cars, how many people drive drunk? If there are as many drunk drivers as there are sober drivers, then — yes — sober drivers are twice as dangerous. If not, though…

Here’s an example: out of 100 drivers, 10 are driving drunk. There are 12 accidents, four caused by drunk people, and eight by sober people. That’s 2/3 of the crashes from sober folks, 1/3 from people who were driving drunk; but that’s 40% of the drunk drivers (4/10) and 8.9% (8/90) of the sober drivers.

(3) Those dangerous Virgos…

Look back at the chart; have you ever heard of Ophiuchus? Look at the International Astronomical Union table and see if that helps you solve the puzzle.

This is an issue of data normalization. Or, restated: the thirteen listed zodiac signs do not have equal periods, because the earth does not “pass through” each constellation with equal time in the sky. The number of days in Virgo is more than double those in Cancer; the number of days in Cancer is almost quadruple the number of days in Scorpio. Look through Kate Follette’s full write-up for the numbers.

(4) Earth vs. basketball

We’re back to the concept of “relative to what” — the bumps on a basketball are much higher relative to the surface area of a basketball than the Himalayas are relative to the surface area of the Earth. (Sorry, Everest…)

(5) Series and diagrams

I’ll let the pictures speak first.

Start here: diagram1

Add a vertical line. See anything new?


Add a bit of shading. See anything familiar?


…these are the numbers 1 through 5, mirrored vertically.

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