Fortifying Your Kingdom: A Strategic Riddle of Walls and Sieges

Imagine your kingdom, a sanctuary for the world's last herd of tiny dino creatures, now threatened by a formidable alliance. Three neighboring nation-states, driven by hunger, are poised to breach your walls and devour your precious herd. With siege weapons on their way, your kingdom's survival hinges on a single, crucial decision: What wall length will save your kingdom?

The Looming Threat: An Alliance of the Hungry

Your kingdom's fortifications can only hold off the encroaching armies for so long. The arrival of siege weapons spells imminent doom. However, you possess a wall fabricator, capable of reinforcing your border overnight. The catch? It can only produce wall segments of a specific, whole number size, which you must determine in advance.

Understanding the Enemy's Arsenal

Intelligence reports reveal that each rival kingdom wields wall-busters of a unique size:

• The Clowns: 6 meters
• The Royals: 9 meters
• The Redheads: 20 meters

These wall-busters can be combined to level wall segments of matching sizes. For instance, two 6-meter wall-busters can destroy a 12-meter wall, while a 6-meter and a 9-meter can break a 15-meter wall. However, a 7-meter wall would remain unscathed.

The Challenge: Maximizing Wall Strength

Your fabricator's slow production rate necessitates choosing the longest wall segment that can withstand any combination of the enemy's siege weapons. What length will render your fortress impregnable?

The Sieve of Eratosthenes: A Timeless Strategy

To solve this complex problem, we can draw inspiration from the Sieve of Eratosthenes, an ancient technique for identifying prime numbers. Devised by the 3rd century BCE mathematician Eratosthenes of Cyrene, this method efficiently eliminates composite numbers, leaving only primes behind.

Applying the Sieve to Wall Construction

We can adapt this technique to our wall problem by strategically eliminating vulnerable wall lengths. By organizing numbers into columns of 6, we can quickly identify and discard entire groups of numbers susceptible to the wall-busters.

Since the enemies can combine their weapons, creating a wall of 15 meters (6 + 9) is possible. This means that if we identify a number vulnerable to the wall-busters, then all the rest of the column below it would also fall. Because your enemies can make 9, they can make 15, 21, 27, and so on by adding the clowns’ 6-meter machines.

Finding the Unbreakable Length

Focusing on combinations of 20s and 9s, we can eliminate even more options. For example, 20 + 9 and 20 + 20, and everything below, can be easily created by your rivals.

In the remaining column, we find that 43 is the largest wall segment that cannot be destroyed by any combination of the siege weapons.

The Solution: Fortification Achieved

By setting your wall fabricator to produce 43-meter segments, you ensure that your fortress remains impenetrable. As the sun rises, your kingdom is safe, and the tiny dino creatures are spared from becoming the enemy's next meal.

This strategic riddle demonstrates the power of mathematical thinking in overcoming seemingly insurmountable challenges. By understanding the enemy's capabilities and applying clever problem-solving techniques, you can safeguard your kingdom and its precious inhabitants.