Answers (go back to puzzles here):

The first batch are from the Cognitive Reflection Test, designed in 2005. (The original paper with analysis of what the test says about your answers is here.)

3 easy puzzles

1. A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost?

The intuitive answer is 10 cents, but the correct answer is 5 cents. (The ball costs 5 cents, and the bat costs a dollar more, or $1.05. $1.05 for the bat + $0.05 for the ball = $1.10 total.)

2. If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?

The intuitive answer is 100 minutes, but the correct answer is 5 minutes. (If 5 machines make 5 widgets in 5 minutes, each machine must make 1 widget every 5 minutes. Therefore, 100 machines would make 100 widgets in 5 minutes.)

3. In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?

The intuitive answer is 24 days, but the correct answer 47 days. (Since the patch doubles every day, it would only take one day to go from half covered to fully covered.)

If you did poorly, you were relying too much on system 1 (fast and intuitive) thinking rather than system 2 thinking (deliberative and logical). Poor results on this test correlate with religious thinking (read a scholarly analysis here).

Each type of thinking has its place, though either one misapplied causes problems. This test highlights the problem with too much reliance on system 1 thinking, but too much system 2 is a problem, too. For example, you could stop and puzzle over whether that noise in the bushes is a squirrel or a predator.

3 more, about as easy

Let’s move on to the next batch of questions (source):

4. If you’re running a race and you pass the person in second place, what place are you in?

You’re in first place, right? No, you’re in second place since you’ve simply replaced the person in the second-place spot. To be in first place, you’d have to pass the first-place person.

5. A farmer had 15 sheep and all but 8 died. How many are left?

No, not 7. If “all but 8 died,” then there must be 8 left.

6. Emily’s father has three daughters. The first two are named April and May. What is the third daughter’s name?

No, not June. If there 3 daughters and the first two are April and May, the third daughter must have the only other name we’re given, Emily. (And, yes, we’re assuming that Emily is a girl.)

The deception in these last two questions is similar to that used in this nursery rhyme puzzle, the earliest forms of which date to the 18^th century.

As I was going to St. Ives,
I met a man with seven wives,
Each wife had seven sacks,
Each sack had seven cats,
Each cat had seven kits:
Kits, cats, sacks, and wives,
How many were there going to St. Ives?

The answer is 1.

The last 3, slightly harder

7. “You are shown a set of four cards placed on a table, each of which has a number on one side and a colored patch on the other side. The visible faces of the cards show 3, 8, red and brown. Which card(s) must you turn over in order to test the truth of the proposition that if a card shows an even number on one face, then its opposite face is red?”

This puzzle is the Wason selection task (1966).

The intuitive answer is the 8 card (I must make sure the other side is red) and the red card (I must make sure the other side is an even number). The correct answer: 8 and brown. The rule we’re testing is, “Even on one side means red on the other.” The red card could have either odd or even on its opposite side; the rule doesn’t care. But we must check the brown card to make sure that its opposite side isn’t even. If it were, that would falsify the rule.

What’s interesting about this puzzle is that it’s isomorphic to a puzzle that most of us find so easy that it’s not even a puzzle. Change the proposition to: “If you’re drinking alcohol, you must be 21 years old,” and have the cards show drinkers’ ages on one side and their drink on the other. The cards you see are 16, 25, Coke can, and beer glass. Obviously, you turn over the 16 (“Let’s make sure you’re just drinking soda”) and the beer (“You’d better be 21”) (Source).

8. You have a rope around an earth-sized sphere at the equator. That makes it about 40,075,160 meters in length. Now you increase the rope length by 10 meters uniformly around the globe so that it’s suspended over the surface equally. What is the gap between the surface of the sphere and the rope?

That’s a really long rope. What difference would 10 more meters make? The intuitive answer would imagine the rope barely less snug around the equator, but the actual gap would be enough to walk under. You’re increased the radius of the rope circle by 10/2π = 1.59 meters, and that’s true whether the original sphere is the earth or a basketball.

9. You have 100 pounds of potatoes, which are 99% water by weight. You let them dehydrate until they’re 98% water. Now how much do they weigh? (Source)

If you’re dropping the water from 99% to 98%, that can’t reduce the total weight by that much, right? Wrong. The answer is 50 pounds.

It’s clearer if you focus on solids, not water. 100 pounds of potatoes = 99 pounds water and 1 pound solids. Remove 50 pounds of water, and now you have 50 pounds of potatoes formed from 49 pounds water and 1 pound solids. Your 50 pounds of shriveled potatoes are now 98% water (49 pounds water/50 pounds total × 100%).

Imagine a variation of the puzzle where your 100 pounds of potatoes were 99.9% water and you wanted them to be 99.8%. To drop that teeny amount, you’re doubling the fraction of solids, which means that, again, you must halve the total weight, and the answer is again 50 pounds.

Got it? Test yourself on this Ask Marilyn column with this variation: “Say you have 200 fish in an aquarium, and 99% of them are guppies. You want to reduce the proportion of guppies to exactly 98%. How many should you remove?”

Study with politics as a variable

One interesting study took a tricky statistical problem but framed it in two ways. The exact same data was given to one set of subjects as a study on skin cream effectiveness (politically neutral) and to another set as a study on concealed handguns (politically charged). The smartest Democrats and Republicans performed equally well, but on the handgun version, they were motivated to (and had the skills to) dig into the problem thoroughly and find the correct answer but more so when it offended their political beliefs about handguns.

♦

Remember today’s lesson, kids: the next time you think that your brain is reliable, remember which organ is telling you this.