Allora and the Puzzles of Archimedes [PDF + eBook + Worksheets]

A fantasy novel that teaches friction, levers and pulleys inside the story — 197 pages, fourteen chapters, for readers ages 9–12. EPUB and PDF, with a puzzle book and three worksheets.

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Allora has turned off friction. All of it.

She's an apprentice to Wizard Leibniz, and the thing she was experimenting with is a real 2,000-year-old puzzle called the Ostomachion — fourteen pieces, attributed to Archimedes, sometimes called the first puzzle ever made. Now nothing grips anything, nails slide out of walls, and the mess is hers to fix. Sorting it out will take Archimedes himself, a Fairy Queen, a band of trolls, and a working understanding of how the world holds itself together.

Written by Sarah Allen, illustrated in black and white by Marie Delwart.

What's inside the book

Fourteen chapters of fantasy adventure — roughly 185 pages — through a world where the physics has been broken and has to be put back.

Twelve "A Note from the Librarian" sidebars, set where the story raises the question rather than collected at the end. Between them they cover static and kinetic friction, Newton's First and Third Laws, why nails hold in wood and what happens when they stop, why lifting the puzzle pieces with a straw works but is harder than it looks — the same trick used to lift panes of window glass in real life — torque and why a doorknob turns, the three classes of lever and how to spot the load, fulcrum and effort in each, and the trick for adding up an arithmetic sequence in your head. Two more are about the people: the real Archimedes behind the character, and Euler. There's also a glimpse of the discovery Archimedes was proudest of, comparing the volume of a cylinder to a sphere.

Things to try, in the middle of the story — sliding your palm flat across a desk to feel what friction actually is, sitting in a chair that spins with your feet off the floor to find out what moving without it would be like, and inventing your own arithmetic sequences to test the summing trick against adding them up by hand.

The Ostomachion itself, which is a real puzzle you can cut out and try. There's a free printable, and some videos about pulleys, at MathwithSarah.com/Allora.

Plus a bonus: Archimedes' Puzzle Book, a 28-page PDF

A separate illustrated puzzle book with a full answer key, built to let a reader practice what Allora learns:

• Lever puzzles — Which Way Will the Lever Tip? · Identify the Effort, Load, and Fulcrum · A Castle Demolition Puzzle
• Pulley puzzles — tension forces in single and compound pulley systems · "Would this actually do anything???", on which pulley contraptions give a mechanical advantage and which don't
• A Certificate of Puzzle Mastery at the end, for anyone who makes it through

Plus three worksheet PDFs, with answer keys

Built as a progression from intuition to calculation to puzzle-solving:

• Friction Explorer — pairs with a free interactive friction simulator on the website. Predict first, then drag a block around, then switch static and kinetic friction on and off to see what really happens. Conceptual, no calculations.
• Calculating Friction — works through the friction equation f = μN with characters from the book, including multi-step problems where the normal force changes: a hawk lifting a chair, pushing down on a broom.
• Flower Garden Riddles — integer puzzles based on Archimedes' Cattle of the Sun problem, introducing primes, squares, triangular numbers and Diophantine equations through trolls, fairies, dwarves and dragons. The original Greek riddle is at the back for the very ambitious.

The math note. The novel itself carries no equations — the physics is explained in words and pictures, and a reader who wants none of the math can have none of it. The calculation lives in the worksheets, where it's optional by design. If you want the algebra, it's the Calculating Friction worksheet you're looking for.

What you'll get

• An EPUB of the book — for Kindle, Apple Books, Kobo and other e-readers
• A PDF of the book — for any computer, tablet or phone, or for printing
• A PDF of Archimedes' Puzzle Book, with its answer key
• PDFs of the three worksheets — Friction Explorer, Calculating Friction and Flower Garden Riddles — each with an answer key

All of them are ready the moment you check out — the download buttons are on the confirmation page. Your order confirmation email also has a View Order and Downloads button, which takes you back to your order page to download them any time.

Looking for the paperback? Allora is also sold as a printed book, which comes with these same files. This page is the digital edition on its own.

Part of the Fairy Tale Physics series

Every book in the series is self-contained, so you can start anywhere. Allora's story stands on its own, though readers of the others will notice familiar faces — Newton and Leibniz both turn up. It isn't part of the Books 1–4 collection, so owning the omnibus doesn't mean owning this one.

About the author

Sarah Allen tutored math and physics for twenty years before becoming a full-time writer. She has a physics degree from the University of Washington (graduated with college and departmental honors) and a master's in Cognition and Learning from Columbia. Her goal is to write the sort of physics books she would have wanted as a kid.

Ages 9–12 · 197-page novel · 28-page puzzle book PDF · 3 worksheet PDFs · Black-and-white illustrations · EPUB and PDF