World Mathematics Championships · Junior
Hard practice for Cycle Solve and Crypto.
All problems are original and every answer has been checked by computer. They're meant to be hard: on most of them you'll get stuck at first, and getting unstuck is the skill being trained. No calculator. Use paper, a pen and a ruler, just like the real round.
Cycle Solve
Each problem uses the previous answer, and the last feeds back into the first. The hard part is finding where to break into the loop. Write up your reasoning; that's what earns marks.
Online format · endlessCrypto · Mystery Operations
Three secret operations ◆ ▲ ●. Test them on the built-in calculator, work out what they do, then combine four numbers to hit each target. The operations change every 20 minutes.
Regional format · 6 packsCrypto · Cipher Packs
Learn a cipher, teach it to a teammate (or a parent) from memory, decrypt two maths questions, solve them, and give your answer encrypted.
How to train with this
- Do it under real conditions. Start the 60-minute timer and use no calculator. Don't open hints until you've been stuck for at least 10 minutes.
- Write the solution, not just the answer. WMC marks the write-up: how you found a way in, why the loop closes, and how you checked it.
- Hints come in steps. Each one gives away a little more. Open them one at a time.
- Mark yourself honestly using the rubric. Then read the model solution and compare the method, not just the number.
- After a few cycles, craft your own. Golden Ticket holders build a Cycle for their interview, so practise that too.
Progress and write-ups are saved in this browser only. This is an independent practice site, not affiliated with WMC; see wmc.app for the official rounds.
Crypto · Online format
Mystery Operations
Each set has three secret operations. Probe them on the calculator, work out what they do, then use all four numbers exactly once (any operations, repeats allowed, any brackets) to make each target. There is no order of operations: without brackets, work left to right.
Crypto · Regional team format
Cipher Packs
How the real round works: each of the three team members gets one pack. You read your cipher, then put the pack away and teach it out loud to your teammates. Then everyone decrypts their own questions, solves them, and hands in the answer written in words and encrypted (e.g. 42 → FORTYTWO, then encrypted). You have 90 minutes in total.
At home: take three packs, teach each one to a parent or friend without looking, then work the questions. The ciphertexts stay the same between visits, so you can come back to a pack later.
Interview preparation
Craft your own Cycle
Golden Ticket holders create their own Cycle round and send it to their mentor before the interview. Crafting a loop is the best way to understand how they work, and it's hard to do well.
- Pick the answers first. Choose the final values (for example A = 8, B = 510, C = 22) before writing any problems.
- Write each problem as a function of the previous answer. "Let A be the answer to Problem 1…" Each problem must make sense for any value it might receive, not just the intended one.
- Build in a way in. Choose at least one: an answer that doesn't depend on its input (e.g. the boarding pass problem); a condition that only one value can satisfy (e.g. the 60° triangle); a small set of cases (e.g. last two digits of powers); or a growth argument that leaves only a few values to check.
- Prove the loop has exactly one solution. If two values both close the loop, the cycle is broken. Write the uniqueness argument down; your mentor will ask about it.
- Mix the topics. A good cycle crosses number theory, geometry, combinatorics and probability, so different team members get a turn to lead.
- Test it on someone. Time them. If they finish in 5 minutes it's too easy; if they never find a way in, add a hint to the wording.
Your draft
Saved in this browser.