Now instead lets divide each square diagonally from the top left corner to the bottom right corner. What is the fewest amount of yellow colorings needed?

]]>The next such arrangement, for which there is exactly 50% chance of taking two blue discs at random, is a box containing eighty-five blue discs and thirty-five red discs.

By finding the first arrangement to contain over 10^{12} = 1,000,000,000,000 discs in total, determine the number of blue discs that the box would contain.

Two people wish to cross a river. There is only one boat. The boat can only carry one person at a time. A person can not cross the river unless in the boat. The boat can not cross the river without someone in it. Both people cross the river. Can you explain how it happened?

]]>How much food did B and C get?

]]>You have a bag of gold coins. They are all of the exact same weight except one, which is a fake.

You have a balance scale, which you can only use N times. What is the maximum number of coins from which you can pick out the fake?

]]>Luckily, the five other prisoners each have phones (no speakerphones) and you have a list of three phone numbers. Two of them call US bases who know the name of the traitor. The other calls a former US base that was infiltrated by the Taliban who could say anything (even changing what they say).

You have enough time to make two rounds of calls, and have enough batteries (that can be passed between the cells) to make six total calls. How can you pull it off?

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You are tasked with designing a robot to explore a large but finite maze. The maze is drawn on a square grid, and walls exist on some of the edges of the square grid. Some of the squares contain a flag.

Your robot may interact with the world in the following ways:

1) Check which of the 4 adjacent edges contain walls.

2) Move to one of the 4 adjacent squares (provided there is no wall in the way).

3) Check if there is a flag on your square.

4) Pick up a flag (provided there is a flag on your square and the robot is not already holding a flag).

5) Put down a flag (provided the robot is holding a flag and there is not already a flag on your square).

6) Generate a random bit.

7) Output a number.

Your robot will be placed in a maze. The maze will contain some number of flags (from 100 to 1000). All flags will be reachable from the robot’s starting position. Your robot is tasked with determining the number of flags. The robot may take as long as it needs, but may only output one number and must output the correct answer eventually, with probability 1.

The catch is that your robot is not Turing complete. It only has a finite amount of memory. You can give your robot as much memory as you need, but it must succeed on arbitrarily large mazes

]]>How much food did A and D get?

]]>]]>

For each generation,

all happy bugs birth 1 sad and 1 neutral bug.

all sad bugs birth 2 happy bugs

all neutral bugs birth one happy bug and one sad bug

all birthing bugs die after birthing. No bug lives more than one generation and birth at the same time. These bugs do not require mating to reproduce.

The initial generation only consists of one happy bug.

Write an explicit function to determine for the nth generation how many happy bugs there are.

]]>place the digits 1-n where n is 4 such that no consecutive digit of any step value, starting at step value, repeats.

here's an example where n is 2.

0 1 0 2 1 0 2 1 2 0 1 ?

1 2 3 4 5 6 7 8 9 10 11 12

here starting from 1 and going a step of 1, there are no repeats. starting from 2 and going a step of 2, no repeats, and so on. However there is no way to get 12 without repeating.

your task is to find the max value for 4.

]]>176, 184, 186, 190, 192, 194, 198, 200, ?, ?

]]>It is still blowing my mind. It just so happens to be from the same round as the problem I posted earlier.

Problem

In this problem, you have to find the last three digits before the decimal point for the number (3 + √5)^{n}.

Like before, there is a small input and a large input for this problem. For the large input, n can reach up to 2 billion. Thus, in order to calculate the answer in a reasonable amount of time (and precision*), some key mathematical insights are required.

What are they?

**for non-coders: since computers cannot store real numbers with infinite precision, most operations on "floating point numbers" cause a gradual loss in precision. In our case, it would almost definitely result in an incorrect answer.*

For example, 349 + 943 = 1292, 1292 + 2921 = 4213 4213 + 3124 = 7337

That is, 349 took three iterations to arrive at a palindrome. Although no one has proved it yet, it is thought that some numbers, like 196, never produce a palindrome. A number that never forms a palindrome through the reverse and add process is called a Lychrel number.

Due to the theoretical nature of these numbers, and for the purpose of this problem, we shall assume that a number is Lychrel until proven otherwise.

In addition you are given that for every number below ten-thousand, it will either (i) become a palindrome in less than fifty iterations, or, (ii) no one, with all the computing power that exists, has managed so far to map it to a palindrome. In fact, 10677 is the first number to be shown to require over fifty iterations before producing a palindrome: 4668731596684224866951378664 (53 iterations, 28-digits).

Surprisingly, there are palindromic numbers that are themselves Lychrel numbers; the first example is 4994. How many Lychrel numbers are there below ten-thousand?

(Here's a problem for my programmers)

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https://www.flickr.com/photos/153009216@N05/albums/72157681733970625

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***

I've checked for all numbers up to 162, it's true:

81* 12345679= 999999999

172839506*162=27999999972

Is there any simple proof for any integer?

]]>https://www.flickr.com/photos/153009216@N05/albums/72157678397870123

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Ask:What additional data would help better this?

You can view here: https://www.flickr.com/photos/153009216@N05/albums/72157679628458322

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