Brain Teasers 10 - Puzzle Prime

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A Broken Circle

Section: Brain Teasers Category: Mathematics

Difficulty: Expert Tags: Featured

There are N points on a circle. If we draw all the chords connecting these points and no three of them intersect at the same point, in how many parts will the interior of the circle get broken?

For example, when N is equal to 1, 2, 3, 4, and 5, we get 1, 2, 4, 8, and 16 parts respectively.

SOLUTION

The answer, somewhat surprisingly, is not 2ᴺ⁻¹, but 1 + N(N-1)/2 + N(N-1)(N-2)(N-3)/24.

In order to see that, we start with a single sector, the interior of the circle, and keep successively drawing chords. Every time we draw a new chord, we increase the number of parts by 1 and then add 1 extra part for each intersection with previously drawn chords.

Therefore, the total number of parts at the end will be:

1 + the number of the chords + the number of the intersections of the chords

Each chord is determined by its 2 endpoints and therefore the number of chords is N(N-1)/2.

Each intersection is determined by the 4 endpoints of the two intersecting chords and therefore the number of intersections is N(N-1)(N-2)(N-3)/4!.

Unknown Author January 13, 2023

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Elemental Puzzles

Section: Brain Teasers Category: Deduction

Difficulty: Medium

Below each of the following Venn diagrams there are seven tiles consisting of two letters. Place each tile in a different region so that the four tiles in each circle can be rearranged to solve the corresponding clue.

SOLUTION

The solutions are shown below.

Greg Wall January 7, 2023

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Discs, Planets, and Time

Section: Brain Teasers Category: Insight

Difficulty: Medium

Which of the following is the odd one out and why?

CD
Venus
Clock
DVD
Mars
Watch

SOLUTION

The answer is Mars, since it spins counter-clockwise. All the other objects spin (or contain elements that spin) clockwise.

Unknown Author January 4, 2023

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The Bicycle Problem

Section: Brain Teasers Category: Science

Difficulty: Hard Tags: Featured

If you pull straight back on a pedal of a bicycle when it is at its lowest position, will the bicycle move forward or backward?

SOLUTION

The surprising answer is that (usually) the bicycle will move backward.

When a bicycle moves forward, the trajectory its pedal traces with respect to the ground is called a trochoid. Depending on the selected gear of the bicycle, that trochoid could be:

Curtate trochoid (for almost all gears of most bicycles)
Prolate trochoid (if the gear is very low and the bicycle moves slowly)
Common trochoid, a.k.a. cycloid (if the wheels of the bicycle and the pedal spin at identical speeds, practically never happens)

Since we are fixed with respect to the ground, by pulling the pedal backward, we are causing it to move leftward along the trochoid and therefore the bicycle will be moving backward. We note that despite that, the pedal will be moving forward with respect to the bicycle (but not with respect to the ground).

You can see a visual explanation of this puzzle in the video below.

Unknown Author January 1, 2023

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Real Word Gogen Puzzle

Section: Brain Teasers Category: Deduction

Difficulty: Advanced

Add the 25 letters between A and Y to the grid below. Each one should appear exactly once. When you have finished, you must be able to spell the following words moving horizontally, vertically or diagonally around the grid:

AXIOM, BANQUET, CHOMP, FAWN, JOY, LURKED, RUN, STY, VYING

SOLUTION

The solution is shown below:

Andrew Reid December 28, 2022

3 Comments

We Go on Vacation

Section: Brain Teasers Category: Practical

Difficulty: Advanced

For this puzzle/game, you will need to find a group of friends, preferably 5 or more. The premise is that you will go together on a vacation but each of you can bring only specific items there. The rules regarding which items can be brought and which not are known by one of the players and the other ones are trying to guess them.

In the exchange below, George is the one organizing the trip and the one who knows which items the rest are allowed to bring.

GEORGE: I will take my guitar with me. What do you want to take?

SAM: Can I take an umbrella with me?

GEORGE: No, you cannot take an umbrella, but you can take some sunscreen.

HELLEN: Can I take a scarf with me?

GEORGE: No, you cannot take a scarf, but you can take a hat.

MONICA: Can I take a dress with me?

GEORGE: No, you cannot take a dress, but you can take some makeup.

Can you guess what the rules of the game are?

SOLUTION

Everyone is allowed to take with themselves only items whose first letter is the same as the first letter of their names. Thus, George can take a Guitar, Sam can take Sunscreen, Hellen can take a Hat, and Monica can take Makeup.

Unknown Author December 22, 2022

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Leave No Squares

Section: Brain Teasers Category: Deduction

Difficulty: Hard Tags: Matchsticks

How many matchsticks do you need to remove so that no squares of any size remain?

SOLUTION

Nine matchsticks are enough, as seen from the solution below.

To see that eight matchsticks are not enough, notice that removing an inner matchstick reduces the number of 1×1 squares at most by 2. Since there are 16 such small squares, in order to get rid of them all, we need to remove only inner matchsticks. However, in this case, the large 4×4 square will remain.

Unknown Author December 19, 2022

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The Die Game

Section: Brain Teasers Category: Mathematics

Difficulty: Advanced Tags: Probability

You pick a number between 1 and 6 and keep throwing a die until you get it. Does it matter which number you pick for maximizing the total sum of the numbers in the resulting sequence?

In the example below, the picked number is 6 and the total sum of the numbers in the resulting sequence is 35.

SOLUTION

No matter what number you pick, the expected value of each throw is the average of the numbers from 1 to 6 which is 3.5. The choice of the number also does not affect the odds for the number of throws until the game ends, which is 6. Therefore, the total sum is always 3.5 × 6 = 21 on average, regardless of the chosen number.

Unknown Author December 7, 2022

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Hexado Puzzles

Section: Brain Teasers Category: Deduction

Difficulty: Advanced

Place arrows along the hexagon edges so that the number of arrows pointing to each hexagon equals the number of dots inside, adhering to the following rules: