Seth Godin recently posted a wonderful brainteaser in Not so good at math:

Let’s say your goal is to reduce gasoline consumption.

And let’s say there are only two kinds of cars in the world. Half of them are Suburbans that get 10 miles to the gallon and half are Priuses that get 50.

If we assume that all the cars drive the same number of miles, which would be a better investment:

• Get new tires for all the Suburbans and increase their mileage a bit to 13 miles per gallon.
• Replace all the Priuses and rewire them to get 100 miles per gallon (doubling their average!)

Would you believe that you save more gas by putting new tires on the Suburbans? Because that’s the right answer.

What’s great about this problem is that it seems so simple, but the result is so astonishing. Even after you know the answer, it’s still hard to get your head around it.

Seth’s point was that we’re not wired for arithmetic. True, but I think what this problem really shows is that we’re wired for making faulty assumptions about numbers. It’s not our arithmetic that fails us in this case. It’s our logic and intuition that do.

How to solve it logically

In 6 Ways To Improve Your Telecommunication, Zack Grossbart shows a simple way to solve the problem by using pictures and plugging in real numbers. He crunches the numbers for one Suburban, one Prius, and a specific number of miles for the commute.

That’s one way I might have done it. Another way I might have done it is by flipping the miles per gallon (1/mpg) to get gallons per mile. When you want to see how much gas you’re burning, the relevant metric is gallons.

	Suburban	Prius
Before upgrade	0.100 gpm	0.020 gpm
After upgrade	0.077 gpm	0.010 gpm
Gas saved	0.023 gpm	0.010 gpm

As you can see, upgrading a Suburban saves 2.3 times as much gas.

How to solve it intuitively

OK, we know how to arrive at the answer. But how can we resolve the paradox? How can getting 30% more mpg possibly be better than getting 100% more mpg? Here are two ways to understand it intuitively.

1. Consider a more extreme version of the problem.

You own a Hummer that gets 5 miles per gallon. You also own a futuristic supercar that can drive across the country on a single drop of gas. Would you rather get 1% better mileage on your Hummer, or 1,000,000% better mileage on your futuristic supercar?

Don’t turn this into a problem of comparing one percentage against another. There’s no point in upgrading the supercar. The percentage you improve it by is irrelevant, because nanodrops of gas don’t matter. But any improvement on the Hummer is huge because it burns a lot of gas.

2. Consider a reworded version of the problem.

All the Suburbans in the world burn 83% of the gas. All the Priuses in the world burn 17% of the gas. Which model should you upgrade?

When it’s phrased this way, the answer is obvious. The wording of the original problem distracted you from what was really important. Of course, problems aren’t always nice enough to phrase themselves in the way that is most convenient for you.

Why doesn’t common sense work?

There are several reasons why it’s so easy to be led astray.

1. We fail to spell out our objective. The wording is critically important, because it’s easy to solve the wrong problem.

Are we trying to maximize the mpg of the average car? No. (If we were, we should upgrade the Prius.) Our goal is to minimize the total amount of gas burned by all cars. So focus on that.

2. It might seem strange that we’re not trying to maximize the average miles per gallon. Isn’t that the same as reducing the total amount of gas burned? Well, it would be, if there was only one car. But averages can be tricky.

If we upgrade the Suburbans, the average car would get (13 + 50) / 2 = 31.5 miles per gallon.

If we upgrade the Priuses, the average car would get (10 + 100) / 2 = 55 miles per gallon.

55 is more than 31.5, so upgrading the Prius means burning less gas, right? It might seem like it should work that way, but there is no mathematical law that says so.

Looking at the average isn’t enough – you need to look at the distribution. Here are three pairs of cars, each pair averaging 10 mpg. The greater the variance within each pair, the more gas is needed to drive a fixed distance.

	Mileage	Gas needed to drive 100 miles
Car 1a	10 mpg	10 gallons
Car 1b	10 mpg	10 gallons
Total gas		20 gallons
Car 2a	5 mpg	20 gallons
Car 2b	15 mpg	6.67 gallons
Total gas		26.67 gallons
Car 3a	0 mpg	infinity gallons
Car 3b	20 mpg	5 gallons
Total gas		infinity gallons

That’s what we get when we keep the distance fixed and look at how much gas we need, which is what we have to do for this problem. But just for fun, let’s keep the amount of gas fixed and look at how far we can drive with the same cars.

	Mileage	Distance driven on a 10 gallon tank
Car 1a	10 mpg	100 miles
Car 1b	10 mpg	100 miles
Total distance		200 miles
Car 2a	5 mpg	50 miles
Car 2b	15 mpg	150 miles
Total distance		200 miles
Car 3a	0 mpg	0 miles
Car 3b	20 mpg	200 miles
Total distance		200 miles

Is that surprising?

3. Reciprocals (mpg vs. gpm) are confusing. We’re trained to think in terms of miles per gallon. But gallons per mile is actually a much more natural unit to work with when you’re looking at how much gas you’re burning.

The Suburban gets 10 mpg, and the Prius gets 50 mpg. The Prius gets 400% better mileage (mpg), but it burns 80% less gas (gpm), so you have to be really clear on what you’re talking about.

The differences between reciprocals get more pronounced when you approach a singularity. 0/1 is very different from 1/0, which is what caused the difference in the previous two charts.

4. We’re told that there are equal numbers of Suburbans and Priuses, and we subconsciously think they should therefore be treated equally. But we need to discriminate. The Suburbans are burning 83% of the gas in the world, so they need to be given more weight. It doesn’t matter how many there are, only how much gas they’re all burning.

Of course, what people should really be doing is trading in their Suburbans for Priuses.

• Slow And Steady Wins The Race
• Logic Test From My Interview
• Einstein’s Riddle (The Zebra Puzzle)
• The Inductive Oracle, The Deductive Merovingian
• Gut Vs. Brain: The Body’s Best Decision-Making Organ
• Go And Reasoning: An Interview With Milton Bradley
• The Illusion Of Certainty
• The Case Of The Stolen Diamond
• The Birthday That Never Arrives
• Everything’s An Illusion: A Glitch In The Matrix