# Difficulty – Medium

Medium difficulty brain teasers and puzzles

## Double the Amount You Need to Do

At the beginning of January, you set a goal to work every day, to reach a total of 5000 minutes of work by the end of the month. But to give yourself a better shot of achieving this, you decide to front-load it—at the beginning of each day, you figure out how much you’d need to work on average on each remaining day to achieve your goal, and then you work double the amount you need to do. For example, if you had 50 minutes left and 5 days left, you would need to work 10 minutes/day, so you would choose to work 20 minutes on that 5th-to-last day.

If you chose to work this way, how long does it take you to complete your goal of 5000 minutes?

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## Stranded in the Wilderness Brain Teaser

A man is stranded in the wilderness, in a remote northern area. There’s a lake nearby, and utility poles carrying electricity, presumably to a nearby town. However, there’s no way he can make it to the town in the freezing cold weather.

The man has a dinghy, two paddles, and an axe, but no devices that can communicate with anyone and no way to make a fire.

How does the man manage to get rescued as quickly as possible?

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## Coin Rotation Paradox

The coin rotation paradox is a famous math problem with an unintuitive solution:

If you roll a coin around the edge of another coin of the same size, from an external perspective, how many rotations does the coin make by the time it returns to its original position?

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## Giant Cat Army Riddle from TED-Ed

In Dan Finkel’s TED-Ed video, he shares this math puzzle, paraphrased as follows:

Dr. Schrödinger is creating an army of giant cats for villainous purposes. Your team of secret agents has located his lab, but needs to get through his unusual security system.

The system displays a single number, and has three buttons that control this number:

• Add 5
• Add 7
• Take the square root of the displayed number

Goal: make the numbers 2, 10, and 14 show on the display, in that order.

Rules:

• The display starts at 0.
• It’s fine if other numbers are displayed in between 2, 10, and 14, as long as they appear in that order.
• The system will malfunction if any number is displayed more than once.
• The system will malfunction if any number greater than 60 is displayed.
• The system will malfunction if any fraction/decimal is displayed.

How can you achieve this?

(Sorry, the giant cat army has nothing to do with the puzzle.)

Source: Dan Finkel and TED-Ed

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## Boy or Girl Paradox

The “boy or girl paradox” is a well-known brain teaser by famous puzzle-maker Martin Gardner. It’s popularly considered a “paradox” because (1) it has a highly unintuitive solution, and (2) its ambiguous wording meant either of two solutions could be valid solutions.

This is a rewording of that brain teaser to eliminate some ambiguity from that original question:

Out of all families with exactly two children, we randomly pick one family that has at least one boy. What is the probability that both children in this family are boys?

Assume only for the purposes of this puzzle that a child can only be a boy or a girl, and that either possibility is equally likely.

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## Two Envelopes Paradox

The two envelopes paradox is a famous brain teaser of sorts, and not a true paradox. The problem is generally posed like this:

You are given a choice between two identical envelopes. One envelope contains some amount of money, and the other contains twice that amount of money. There is no way to distinguish between the two. However, when you choose one of the envelopes, before opening it, you are given the option of switching to the other envelope. Should you switch?

Why is this sometimes called a paradox? Well, if you choose to switch, you have a 50% chance of doubling your money, and a 50% chance of halving your money. If the amount of money in the envelope you initially chose is M, this reasoning suggests the expected amount in the other envelope is (2M + 0.5M) / 2 = 1.25M. This is more than M, so you should always switch.

But that would suggest once you’ve switched, you’re in the same position you were before you switched, so you should switch again. What is the problem with this reasoning?

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## Honkai Star Rail Dissatisfied Cycrane Truth and Lie Riddle

In Honkai: Star Rail, one side quest takes you to solve a number of puzzles involving cycranes, including one fun and tricky Dissatisfied Cycrane truth and lie riddle!

“I either can only tell the truth, can only tell lies, or must say a lie after a truth. You can only ask two questions, and then I’ll ask for your answer.”

—Dissatisfied Cycrane
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## Ministry of Education Quiz

Honkai: Star Rail is the latest mobile game by the creators of Genshin Impact, and shortly into your gameplay, you’ll run into the Ministry of Education quiz as part of your daily quest. Over the course of multiple daily quests, you’ll run into a mix of math and logic puzzles – here are the Honkai Star Rail Ministry of Education quiz answers, with explanations!

## Honkai Star Rail Ministry of Education Quiz Answers

### Part 1 Question 1

Which are there more of: prime numbers, or natural numbers?

• Prime numbers.
• Natural numbers.
• The same.
• What are prime numbers?

#### Click to show answer to Part 1 Question 1

The same.

To answer this, you need to apply the principles of set theory, which is usually taught in university level mathematics classes. In short, both prime numbers and natural numbers are infinite (but some infinities can be bigger than others), and infinite in a way that every prime number can be matched to exactly one natural number (1st prime number is matched to 1, 2nd prime number is matched to 2, etc.).

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