Category: Algebra

Recursive Renewals

You will need to do algebra, combinatorics, calculus, and trigonometry, and also use multiple matrices, to solve today’s challenge.

When you take a book out at the Leibniz Library, you must return it before the fine equation, f(x), rises above zero. If not, your fine is determined by the fine equation. The variable r represents the number of times you have renewed the book, beggining at zero. The variable d counts days and increases by 1 every day, beggining at zero. The variable t counts days and increases by 1 each day, beggining at 0. Each time you renew the book, the variable r in the fine equation, which begins at zero, increases by 1, and the variable t is reset to 0. You may not renew the book if someone else has it on hold or the fine equation is above 0. The probability that someone puts your book on hold on any given day is determined by the hold equation, h(x). How can you maximize the number of days you have with your book without paying anything?

f(x)=d^2+5d-7cos(r)+r^3-100\\[10pt] h(x)=\frac{t^2+3sin(t)+15}{100}
Solution

Ridiculous Radical

This challenge covers the topics of calculus and algebra.

In calculus we often find the derivatives of things. The derivitave of 1, π, of any other constant c, is 0. The derivative of x is 1, and that of any constant multiplied by x is that constant. And then the derivative of x2 is 2. All common knowledge. But what about the derivative of √x? That would be √x / 2x, We multiply by the exponent and then decrease it by 1. This does work with x2. But what about the derivative of √x+1? Try to factor the 1 out of the radical. That is today’s challenge. Good luck!

\frac d {dx} \sqrt {x+1}
Solution

Machiavellian Matrix

You will need serious matrix solving skills to solve today’s challenge.

Todays challenge is strait forward: solve the Machiavellian Matrix. Machiavellian means crafty and duplicitous, like a trickster. Sometimes this means breaking the rules or finding loopholes. With matrices, machiavellian often means horrifying. Variables are x, y, and z, sum on the right.

\left(\begin{matrix} xyz & xyz & xyz & xyz \\ x & y & z & 1 \\ 4 & 1 & 1 & 0 \end{matrix}\right)
Solution

Peculiar Pets

You will need matrices and combinatorics to solve this challenge.

In a city with infinite people, there are only three types of pets. They are dogs, cats, and fish. 60% of people have a pet. The ratio of dogs to cats is 2:3, and the ratio of poeple with cats to those with fish is 5:6. All pets are randomly distributed. The challenge is to find what fraction of the people own dogs.

Solution

Fascinating Fruits

Diagram of the Fascinating Fruits.

You will need an understanding of algebra and combinatorics to solve today’s challenge.

Today’s challenge begins with 3 rows of fruits. They begin with an apple, and orange, and a pear. The fruits behind the front are unknown. Directly behind any apple, there is an equal chance of a plum, a banana, or an orange. There is no other fruit directly behind an apple, but one apple is at the end of a row. Behind any plum is the same fruit that was in front of it. Directly behind an orange, there is a one third probability there are grapes. There is an equal probality of pears and blueberries, and 1 orange leads to a banana. For any banana there is an equal chance of blueberries or a plum. 1 banana is followed by grapes, and another ends a row. Behind blueberries, there is a one third chance of a pear, otherwise grapes. Directly behind grapes, there is a 50% chance of more grapes, a 25% chance of a plum, and otherwise a pear. Finally, directly behind a pear there is the fruit of the colour in the rainbow following that of the fruit in front of the pair. The exception is the 1 pear that ends a row. The apples in this question are red, which is considered to follow purple, the colour of the plums. The fruit following the pair at the front is an apple. The challenge is to use this information to find the probability that the row started by a pear ends with one.

Solution

Puzzling Probabilities

This challenge, though it appears simple, is far more difficult than you might expect. Knowledge of algebra, combinatorics, and graph theory, as well as ingenuity and effort, are necessary to successfully complete this challenge. Be warned.

Bob goes to Scam Casino to lose some money. He decides to play the game “Stupid Spin.” There are many equal pieces of the spinner, and only a few give Bob a point. Otherwise, the Casino gets one. The probability that Bob gets a point is a, and the first to b points wins. The challenge is to find the probability that Bob will win a given game of Stupid Spin.

Solution

Crazy Capture

All of your algebra, geometry, trigonometry, angle theorems, and pathmaking skills are required to have a chance of solving today’s challenge.

2D capture example.

In a mathematical capture, all police officers and the target are points that travel at the same constant speed in infinite space in n dimensions. They have infinite endurance and perfect reaction time. The target is instantly captured and cannot ecape when a police officer touches it. In 1 dimension, when n = 1, it obviously takes 2 officers surrounding the target. When n = 2, we can do it with three easily. We do this by dividing the plane into three, each officer can easily trap the target in a 120° slice. The challenge is to see how many officers it takes to capture the target when n equals 3, in three dimensions.

Solution

Devising Deactivation

To solve this challenge, you will need to use geometry, trigonometry, and algebra, to do pathmaking.

Map of the radar that must be deactivated.

You are working for the government as a mathematician and they tell you of a problem. There is a plan afoot to compromise an enemy radar station. It is critical that the invading crew member is undedected, as the enemy must not know that their radar is sending them misinformation. However, the radar has a range of 100 metres and completes a rotation every 12 seconds. The fastest crew member can only sprint at 7 metres per second with their toolkit. Because of the radar’s length and rotation speed, it is nearly impossible to disable it undetected. None of the simple strategies work. The challenge is to find a path that the invader can take without being detected.

Solution

Absurd Algebra

Extreme algebra knowledge and skills are required to solve this challenge.

Absurd Algebraic Graph.

This time, we are faced with a strange graph. When x rounds down to an even number, the slope is 1. But when it rounds down to an odd number, the slope is -1. This makes an intriguing zig-zag pattern with peaks always at 2 or 3. But what equation could possibly give such a line? Well, that is the challenge.

Solution

Captivating Circles 5

You will need to use algebra to solve this challenge. Some programming would be extremely useful as well.

In the video-game “Battle of the Blobs”*, a blob is a circular creature that deals damage over time proportional to its current area. A blob’s area will shrink proportionally to damage dealt to it. The damage over time is not dealt at short intervals, rather is always being dealt, but with the same rate. Our challenge begins when 2 blobs, Jack and Jill, battle each other. The radius of Jack is 1, and the radius of Jill is 1.5. Clearly, Jill will kill Jack. The question is what will be Jill’s radius after fighting Jack, assuming the only change in a blob’s size is caused by damage from the other.

*”Battle of the Blobs” is not a real video-game.

Solution