Today is 17 July 2018, the 198th day of the year.



more quotes

more humor

Die-Cast Polyhedra banner



  Platonic Realms Math Store Banner
In ancient Greek mythology, the hero Heracles (Latin Hercules) is set many seemingly impossible tasks. One of these tasks is to slay the Hydra, a dragon-like monster with many heads and a fiendish reputation. Moreover, when one of its heads is chopped off, two more immediately grow in its place. This presents Heracles with a poser, which he solves by ducking under all the heads and stabbing the creature in its hideous heart. While granting Heracles due regard for “getting to the heart of the matter,” we consider here the more abstract problem of the multiplying heads on its merits.
Let us suppose that any given Hydra has some number of necks sprouting from its body, each of which may "branch out" any number of times, and that each branch may in turn branch again, and so on. At the terminus of each of the final branches is a head, and we suppose that no Hydra has more than finitely many heads. A typical such Hydra is represented by the following diagram:

We'll refer to the points where branchings occur as nodes. Suppose now that you can cut off only one head at a time, and that when you do so it and its neck are eliminated back to its supporting node. However, the neck-segment supporting that node and everything left on it will be replicated on its supporting node. Moreover, it will be replicated a number of times equal to the number of heads you have chopped off. Any head directly supported by the body can also be cut off, but nothing is duplicated in that case.
For example, suppose that the first head you chop off is the dark blue one in the diagram. Then the pale blue necks and heads will be replicated – in this case doubled – in place. If the next head chopped off is the dark red one, then the pink necks and heads will be replicated twice over, resulting in three copies of the pink necks and heads where before there was one. And so on. You may chop heads off in any order you choose, provided you only chop off one head at a time. Unlike Heracles, you may not attack any other part of the Hydra.
Today's challenge – find a strategy to slay the Hydra!


Solution to yesterday's challenge

THE MATH STORE LIVES! At last, the new Math Store is open, including all the downloads we have offered in the past. Please come poke around the shop and let us know what you think!

NEW FOR 2012: Platonic Realms is at last being rewritten and moved to a new server. Please check back soon to see the new look, new content (together with all your old favorites), and an all new math store!


This site is registered with the Internet Content Rating Association (ICRA), and is suitable for all ages. Because we value freedom of expression on the internet as well as the privacy and comfort of our visitors, Platonic Realms is also bug-free and blink-free. Platonic Realms uses cookies for site-traffic analysis, but these automatically expire soon after your visit. Our privacy policy is very simple: we ALWAYS ask our visitors before collecting any personal information, and we NEVER share information about our visitors with anyone else. Period.


Copyright © 1997-2013, Math Academy Online™ / Platonic Realms™. Except where otherwise prohibited, material on this site may be printed for personal classroom use without permission by students and instructors for non-profit, educational purposes only. All other reproduction in whole or in part, including electronic reproduction or redistribution, for any purpose, except by express written agreement is strictly prohibited. Please send comments, corrections, and enquiries using our contact page.