Resolving the Paradox

How to Explain the Unexplainable

A comprehensive guide to finding the single new fact that makes two contradictory statements true.

The Puzzle
The Clue
The "Aha!"

What is Your Mission?

You will be given two facts that seem to contradict each other.

Your mission: Find the one new piece of information that allows *both* facts to be true at the same time.

The "Two-Story House" Analogy

Fact A

"The top floor of the house is burning hot."

Fact B

"The bottom floor of the house is freezing cold."

The Resolution

"The house's heater is broken and stuck *on* upstairs, while the A/C is broken and stuck *on* downstairs."

The resolution doesn't deny the facts—it provides a new piece of information that explains how both can be true.

The Anatomy of a Paradox

You are looking for the "Ah, *that's* why!" moment.

Example: The Water-Saving City

Fact A: "The population of City X grew by 20% last year."
Fact B: "The total water consumption of City X *decreased* by 10%."
The Conflict: "But... I would expect 20% *more* people to use 20% *more* water, not less!"

The Resolution (The Correct Answer):

"Last year, City X mandated that all homes and businesses install new, high-efficiency fixtures that use 30% less water."

(This new fact explains *how* the total could drop, even with more people.)

Decoding the Question Stems

They all ask the same thing: "Find the 'Aha!' moment."

  • "Which of the following, if true, most helps to resolve the apparent paradox?"
  • "Which of the following, if true, most helps to explain the discrepancy?"
  • "Which of the following, if true, reconciles the surprising finding?"
  • "...most helps to explain the surprising phenomenon?"

The Cardinal Rule: Do NOT Deny the Premise

A weakener *attacks* the premises. A paradox-resolver *accepts* them.

The correct answer will *never* question the facts. It adds a *third* fact that makes the first two make sense. Any answer that says one of the facts is wrong is an *invalid* trap.

Example: The Water-Saving City

Paradox: More people, less water.

Invalid Answer (A "Denier"):

"The census data was wrong; the population did not actually grow."

This is a trap because it *denies Fact A*.

Valid Answer (A "Resolver"):

"The city implemented a new, mandatory water-saving program."

This is valid because it *accepts both facts* as true.

Three Common Paradox Patterns

1. The "Opposite Result"

An action was taken to achieve a goal, but the *opposite* of the goal happened.

Example: "The company hired a new CEO to raise profits, but profits fell."

How to Resolve: Find the *unintended consequence* or *cost* of the action. (e.g., "The new CEO's salary was so high it erased all the gains.")

2. The "Two Different Trends"

Two things you'd expect to move together move in opposite directions.

Example: "Car sales are up, but gasoline sales are down."

How to Resolve: Find the *third factor* that explains *both* trends. (e.g., "A new, popular line of highly fuel-efficient cars was released.")

3. The "Total vs. Part"

A total or average moves in one direction, while a part moves in another.

Example: "The company's *average* salary went up, but my *personal* salary went down."

How to Resolve: Look for a *change in composition*. (e.g., "The company fired all its low-wage workers, which raises the *average*.")

The 4-Step Strategy to Victory

1

Isolate Fact A and Fact B. State them clearly in your mind. (e.g., Fact A: "Revenue up." Fact B: "Profits down.")

2

Identify the Conflict. Articulate *why* it's weird. (e.g., "But... I expect profits to *rise* when revenue rises!")

3

Pre-think a "Bridge." What could make *both* true? (e.g., "The only way this works is if *costs* rose even faster than revenue.")

4

Test the Answers. Find the answer choice that matches your pre-think. (e.g., Find the answer that says, "The new revenue came from a product with razor-thin margins and huge shipping costs.")

The Hall of Shame: Common Traps

These wrong answers are designed to confuse you.

1. The "Denier"

The answer says one of the facts is wrong. (e.g., "The census data was mistaken."). This is *never* the right answer.

2. The "One-Sider"

The answer explains Fact A, but does *nothing* to explain Fact B (or vice versa). (e.g., "The population grew because of new jobs." -> This explains Fact A, but *not* why water use *fell*.)

3. The "Deepener"

This answer makes the paradox *even worse* or *more* confusing. (e.g., "The new residents all installed large swimming pools.")

4. The "Irrelevant"

The answer is on-topic, but doesn't connect the two facts. (e.g., "The city also built a new park." -> So? How does this explain water use?)

Practice Set 1

Apply the 4-step strategy.

Passage 1 (4 options): "A new pest-control program in a farming region was a huge success, eliminating 90% of the crop-eating insects. However, in the following year, the *total crop yield* for the region *decreased*."

Which, if true, most helps to resolve the apparent paradox?

Explanation:
Fact A: Pests were eliminated. Fact B: Crop yield *decreased*.
Conflict: I'd expect *more* crops if pests are gone.
(A) is a "One-Sider." It explains *cost*, not *yield*.
(B) and (D) are irrelevant to the paradox.
(C) Correct. This is the "Aha!" moment. The program had an *unintended consequence*. It killed the bugs (Fact A), but those bugs were *needed* for pollination, so the crops failed (Fact B). Both facts are true.

Passage 2 (4 options): "A company-wide survey showed that 90% of employees who work from home are 'highly satisfied.' However, the company's *turnover rate* (the rate at which employees quit) for work-from-home employees is *higher* than the rate for in-office employees."

Which, if true, most helps to explain the surprising finding?

Explanation:
Fact A: Remote employees are happy. Fact B: Remote employees quit more often.
Conflict: I'd expect *happy* employees to *stay*, not quit.
(A) and (C) are irrelevant. (D) might explain future turnover, but not *past* turnover.
(B) Correct. This resolves it. The employees aren't quitting because they are *unhappy* (Fact A is true). They are quitting because their skills are in demand, and it's easy for them to find *even better* jobs. Their high turnover is unrelated to their satisfaction.

Practice Set 2

Passage 3 (5 options): "In the past five years, the number of airlines competing on the popular New York to Chicago route has *doubled*. However, the *average price* of a ticket on this route has *increased*."

Which, if true, most helps to resolve the apparent discrepancy?

Explanation:
Fact A: Competition doubled. Fact B: Price increased.
Conflict: I'd expect *more competition* to lead to *lower prices*, not higher.
(A) and (B) are irrelevant. (C) and (D) are "Deepeners" - they make the paradox *worse*.
(E) Correct. This provides a *third factor*. The forces of competition *are* pushing prices down, but a massive *increase in costs* (fuel) is pushing them up even more. This allows both facts (more competition, higher prices) to be true.

Passage 4 (5 options): "A new drug, 'Cardia,' was shown to be highly effective in reducing cholesterol. However, in the two years since Cardia was released, the number of deaths from heart disease in the country has *risen*."

Which, if true, most helps to explain the surprising result?

Explanation:
Fact A: Drug is effective (reduces cholesterol). Fact B: Heart disease deaths *rose*.
Conflict: I'd expect an effective heart drug to *lower* heart disease deaths.
(A) and (B) are irrelevant. (D) just restates Fact A. (E) is irrelevant.
(C) Correct. This is an "Opposite Result." The drug *does its job* (Fact A), but it has an *unintended consequence* (fatal side effect) that *also* relates to the heart, leading to more deaths (Fact B).

Final Interactive Quiz

Test your mastery. Find the resolution.

1. (4 options) Paradox: A recent study showed that people who drink coffee just before a 20-minute nap feel *more* refreshed and alert upon waking than people who just take a 20-minute nap. Which, if true, most helps to resolve the paradox?

2. (5 options) Paradox: A company's *total revenue* from its software division increased by 10% this year. However, the *average price* per software license sold by the division *decreased* by 20%. Which, if true, most helps to explain the discrepancy?

3. (4 options) Paradox: The deer population in a national park was declining. To help, park rangers imported a number of wolves, the deer's natural predator. In the following years, the deer population *increased*. Which, if true, most resolves the paradox?

4. (5 options) Paradox: A new, highly-addictive video game was released. In the month after its release, sales of *other* video games at the country's largest retailer *also increased* significantly. Which, if true, most helps to explain this?

Summary & Next Steps

Key Takeaways

  • Find the Conflict. Clearly state why the two facts seem contradictory.
  • Never Deny the Facts. The correct answer accepts *both* facts as 100% true.
  • Look for the Third Factor. The answer is a *new piece of information* (a cause, a cost, a side effect) that explains the discrepancy.
  • Beware of "One-Sider" traps that only explain one fact, not both.

Practice Plan

  • Be a Detective: Read news headlines (especially in business or science) and look for paradoxes. (e.g., "Company's stock is down, but profits are up.")
  • Pre-think Relentlessly: For every paradox, pause for 10 seconds and say "This would make sense if..." before looking at the answers.
  • Review Traps: When you get one wrong, identify the trap. Was it a "One-Sider"? Was it a "Denier"? Naming the trap helps you avoid it next time.