Poor Pigs

The problem can be thought of as encoding information using the pigs. By understanding how many times we can test the pigs within the minutesToTest and how many buckets a single pig can identify, we can find the minimum number of pigs needed to identify the poisonous bucket.

We can perform tests = minutesToTest // minutesToDie + 1 tests within the allotted time. If we use one pig, we can identify tests buckets, as the pig can be fed from different buckets in each test. Using more pigs allows us to increase the number of identifiable buckets exponentially.

Python solution:

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class Solution:
    def poorPigs(self, buckets: int, minutesToDie: int, minutesToTest: int) -> int:
        tests = minutesToTest // minutesToDie + 1
        pigs = 0

        while tests ** pigs < buckets:
            pigs += 1

        return pigs

Explanation:

We calculate the number of tests that can be performed within the given time.
We initialize a variable pigs to keep track of the number of pigs needed.
Using a while loop, we check if tests ** pigs is less than buckets. If it is, we increment pigs by 1.
When tests ** pigs is greater than or equal to buckets, we have enough pigs to identify the poisonous bucket, and we return pigs.

The time complexity of this solution is (O(\log \text{{buckets}})), making it efficient for the given constraint.

The key insight is to recognize that the problem is analogous to encoding information, where each pig represents a digit in a base-tests number system, and the number of digits required determines the number of buckets that can be identified.

10 Prerequisite LeetCode Problems

“Poor Pigs” requires combinatorics and mathematical reasoning, logic, and an understanding of base number systems.

Here are 10 problems for a solid foundation in these areas:

Problem 7. Reverse Integer This problem helps you understand number systems and digits manipulation. You need to reverse an integer without using any string conversion functions.
Problem 191. Number of 1 Bits This problem requires understanding of bitwise operations and base 2 numbers. You’re required to write a function to determine the number of ‘1’ bits.
Problem 202. Happy Number This problem helps you understand number systems and digits manipulation. It asks you to determine if a number is “happy”.
Problem 204. Count Primes This problem deals with understanding number theory and prime numbers, which is helpful for reasoning about combinatorics and probabilities.
Problem 231. Power of Two This problem requires an understanding of bitwise operations and base 2 numbers. It asks you to determine if a number is a power of two.
Problem 263. Ugly Number This problem requires you to write a program to check whether a given number is an ugly number, which exercises logic and mathematical reasoning.
Problem 292. Nim Game This is a game theory problem that requires logical reasoning to find a winning strategy.
Problem 326. Power of Three Similar to Problem 231, this problem asks you to determine if a number is a power of three, which aids in understanding base number systems.
Problem 342. Power of Four Like the previous two, this problem requires an understanding of base numbers. It asks you to determine if a number is a power of four.
Problem 400. Nth Digit This problem helps you understand number systems and digits manipulation. You’re required to find the nth digit of the infinite integer sequence.

These cover number theory, logic, base number systems and combinatorics, which is helpful in tackling the “Poor Pigs” problem.

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class Solution:
    def poorPigs(self, buckets: int, minutesToDie: int, minutesToTest: int) -> int:
        return ceil(log(buckets) / log(minutesToTest / minutesToDie + 1));

Problem Classification

The problem requires determining the minimum number of pigs needed to find a poisonous bucket within certain rounds. This is a Minimum Resource Allocation Problem.

Language Agnostic Coding Drills

Basic Arithmetic Operations: Understanding basic mathematical operations like division and addition is crucial to solve this problem.
Logarithms: The problem involves calculating logarithms. Logarithm converts multiplication and division into addition and subtraction, making calculations simpler. The base of the logarithm can be changed to represent different situations.
Ceil Function: The ceil function is used to round up numbers. This function always rounds a number up to the nearest whole number.
Understanding the Problem Statement: This is a logic puzzle, which means it requires reasoning skills to solve. The problem involves determining the minimum number of pigs needed to find a poisonous bucket within a given time frame.

When these individual concepts are understood, they can be combined to solve the problem.

Problem Solving Approach

Start by understanding the problem statement and the constraints.
Think about the relationship between the number of pigs, the number of buckets, and the time it takes to test for the poison.
Realize that each pig represents a dimension in which the buckets can be arranged. Therefore, the problem can be reduced to calculating the dimensions required to represent all the buckets.
Consider that the number of tests a pig can perform is determined by the total time available (minutesToTest) divided by the time it takes for a pig to die from the poison (minutesToDie), plus one (to account for the first test).
Understand that this can be represented mathematically using a logarithmic equation. The base of the logarithm represents the number of tests a pig can perform and the number itself is the number of buckets. The result of this calculation is the number of pigs required.
Realize that we need to round up the result to the nearest whole number, because we can’t have a fraction of a pig. This is where the ceil function comes in.
Combine these insights to write the code. The code calculates the logarithm of the number of buckets (base is the number of tests a pig can perform), then uses the ceil function to round the result up to the nearest whole number. This gives the minimum number of pigs required.

Targeted Drills in Python

Basic Arithmetic Operations:

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# Declare some variables
a = 10
b = 5

# Perform basic arithmetic operations
sum_ab = a + b
diff_ab = a - b
mult_ab = a * b
div_ab = a / b

# Print the results
print("Sum:", sum_ab)
print("Difference:", diff_ab)
print("Multiplication:", mult_ab)
print("Division:", div_ab)

Logarithms:

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# Import the math module
import math

# Declare a number
num = 100

# Calculate the logarithm base 10
log_num = math.log10(num)

# Print the result
print("Logarithm base 10:", log_num)

Ceil Function:

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# Import the math module
import math

# Declare a decimal number
decimal_num = 7.3

# Calculate the ceil of the number
ceil_num = math.ceil(decimal_num)

# Print the result
print("Ceil:", ceil_num)

Understanding the Problem Statement: For this one, as it’s more conceptual and dependent on the problem at hand, we don’t have a specific drill. It is, however, helpful to regularly practice problem-solving questions to build your skills in this area.

Problem Specific Drill:

For the problem at hand, an important concept is understanding how to translate the constraints and requirements of the problem into mathematical terms. This often involves some degree of logical reasoning and may also require knowledge of certain mathematical concepts. Here is a problem-specific drill:

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# Assume we have 100 buckets, it takes 15 minutes for a pig to die from the poison,
# and we have 60 minutes to test.

# The number of tests a pig can perform is given by:
minutesToDie = 15
minutesToTest = 60
tests_per_pig = minutesToTest / minutesToDie + 1
print("Tests per pig:", tests_per_pig)

# Now assume we have 100 buckets. The number of pigs required can be calculated using:
buckets = 100
pigs_required = math.ceil(math.log(buckets) / math.log(tests_per_pig))
print("Pigs required:", pigs_required)