Facilitator guide for numbered wave cards
Updated 14 November 2026 · 64-slip detection model
Make your detector bright and the other detector dark, then send photons until one is caught. Allow about 10 minutes. Use the 40 wave cards, Wave Board, two tokens, an opaque cup, and a stock of 32 blue A slips, 32 purple B slips and 64 gray missed slips. All slips must have identical size, shape and feel. The cup holds exactly 64 slips, selected from this stock. Use identical opaque card backs or sleeves.
The number is wave strength (amplitude): move that many squares in the arrow direction. Blue ↑ is up, red ↓ down, purple → right, gold ← left. Each direction has seven strength-1 and three strength-2 cards. All plots use the same scale and frequency. The arrow describes a board move, not a photon’s travel.
Put both detector tokens at START (brightness 0). Shuffle and deal three cards each. Each player secretly chooses one card and a target detector. Reveal both, resolve A’s card first and then B’s, and draw back to three cards. Apply the full 1- or 2-square move. Keep used cards in a discard pile; reshuffle them if the draw pile runs out.
A player cannot choose a card that would leave the board on the selected detector. If the first revealed move makes the second move leave the board, the second card is lost. If no card is legal on either detector, discard one instead. A 2-square move cannot be shortened to fit.
After eight rounds, put A brightness blue slips and B brightness purple slips in the cup. Add gray slips until the total is exactly 64. Mix and draw without looking. Blue or purple wins. Gray means neither detector caught this photon: REPLACE the gray slip, mix, and send a NEW photon with unchanged settings. Repeat until caught. If both are dark, play one more round instead. The browser shows the first attempt, then advances retries automatically after a miss. New game or Play again cancels pending photons and keeps the selected play mode.
At one square above START, a ↓1 card returns the token to START and makes brightness 0. A ↓2 card instead crosses START and ends one square below it: brightness stays 1. A larger card is not always better.
At position (x, y), the model’s wave is y cos(2πt/T) + x sin(2πt/T), with amplitude √(x² + y²). Brightness is x² + y². Each coordinate runs from -4 to +4, so each detector reaches at most 32 and their sum cannot exceed 64. PER PHOTON: P(A) = A brightness / 64, P(B) = B brightness / 64, and P(missed) = (64 - A brightness - B brightness) / 64. EVENTUAL WIN: with positive total brightness, A wins with A brightness / total brightness and B with B brightness / total brightness.
For brightnesses 25 and 9, put 25 blue, 9 purple and 30 gray slips in the cup. Per photon, the probabilities are 25/64, 9/64 and 30/64. With replacement, among detections the winner probabilities are 25/34 and 9/34. Misses are new trials, not additional turns or lost points. For brightnesses 1 and 0, it takes 64 photons on average to detect one; the browser compresses further misses using the same replacement probabilities. With both dark, detection would never happen, so play another round.
Up begins at a crest, down at a trough, right at zero rising, and left at zero falling. In-step amplitudes add. Opposite phases subtract; complete cancellation requires equal amplitudes. ↑2 + ↓1 leaves ↑1, while ↑2 + ↓2 gives zero. Equal-amplitude waves a quarter-cycle apart combine to an amplitude √2 times either wave. A strength-2 card from START gives brightness 4.
An ideal single-photon source feeds a 50:50 splitter. Each branch has an independently adjustable loss channel and an ideal detector. Set the branch transmissions to A brightness / 32 and B brightness / 32. Multiplying each transmission by the branch probability 1/2 gives exactly A brightness / 64 and B brightness / 64. Undetected photons leave through loss channels. Both detectors can brighten when losses decrease.
Cards add wave contributions on an amplitude board; the calculated brightness sets each branch transmission. This matches the final detection probabilities, not a literal optical path opened by every card. Card moves are not physical gates. Interference can redirect light to other outputs; cancellation alone does not prove absorption. The animation is a probability sample, not a photon trajectory. Real waves can have any phase; the cards use four.
Start with one cancellation example, then let learners play. Ask: When is strength 1 better than strength 2? Why can adding a wave make a detector darker? Why can a good strategy lose one game? What is the difference between one photon and the eventual winner? Show both sets of probabilities alongside the sampled result. In Wave Lab, compare partial cancellation, complete cancellation, and a phase reversal. A simple challenge is to reach a dark detector using a strength-2 wave and smaller opposite waves.
Developed using QCaMP material. The arrow addition model is related to the phasor method described in Feynman’s QED (1985). The wave board and card game are classroom teaching models.