Two mechanisms on paper

Print one copy per pair or table. The activities are untested with teachers, not a settled workshop decision. All times are estimates.

Exercise 1 · A crayon for computing

A computation is changing marks into other marks by following rules exactly. Counting crayons into cups already does it: 3 crayons in one cup, 5 in another, move them one at a time from the second cup to the first, and you have 8. In 1961 Marvin Minsky showed that cups, counters and two rules, add one and take one, are enough for any computation at all. Here the counters are crayons.

The counter machine

Print the memory sheet and step cards from machine-cards.html. Memory is four cups of crayons, numbered 1 to 4. Every box on a step card holds a written number, or STOP. The program is a set of ADD and TAKE cards. Write a number in every box of a card: its own Step number, the cup, and which step comes next, or STOP. The cards can lie in any order. Keep a finger on the current card.

Program 1 · Add

Step 1: TAKE, cup 2, got one → step 2, empty → STOP. Step 2: ADD, cup 1, next step 1. Put 3 crayons in cup 1 and 5 in cup 2: cup 1 ends with 8 after 11 moves. Any two numbers work.

Program 2 · Multiply

Step 1: TAKE cup 1 (→ 2, empty → STOP). Step 2: TAKE cup 2 (→ 3, empty → 5). Step 3: ADD cup 3 (→ 4). Step 4: ADD cup 4 (→ 2). Step 5: TAKE cup 4 (→ 6, empty → 1). Step 6: ADD cup 2 (→ 5). Put 3 in cup 1 and 4 in cup 2: cup 3 ends with 12 after 70 moves.

From two steps to any program

Groups of cards that do one job become new steps: Program 1’s two cards are MOVE cup 2 into cup 1, and Multiply is COPY inside a loop. Building each new step only from ones you already trust, you can climb a ladder: clear, move and copy; add, subtract and compare; multiply, divide and remainder; numbered memory cells packed into one big number; a processor’s instructions (load, store, add, jump, call); and finally any program a computer runs. Nobody would build a computer this way, but nothing more is needed.

Programs for your classroom

Setup pages for each are in machine-cards.html. For older students, homework.html builds a whole processor’s instructions from ADD and TAKE, level by level.

Why the counter machine is Turing complete

Marvin Minsky proved in 1961 that a machine with just ADD and TAKE steps can do anything a Turing machine can, and so anything any computer can, given enough crayons; even two cups are enough in principle.

Discuss: which step needed you to understand the program? What changed between the three programs, and what never changed? Can you write a program that subtracts? How would you know if a program will ever stop?

Facilitator: before the session, prepare per table the printed cards from machine-cards.html, and a box of 24 crayons. Allow about 19 minutes; rehearse Program 2.

Exercise 2 · A word from the hat

Your table writes one sentence together, one word at a time. Tables of 4 to 6 people. You need a hat or cup, small paper slips, pencils and one response sheet. Choose one reader; the reader also writes slips.

The prompt, for every table:

At recess, we found a…

The reader writes the prompt at the top of the response sheet. The words you draw continue that sentence.

Each turn

  1. The reader reads the prompt and every word added so far.
  2. Everyone writes one next word, or STOP, on a slip.
  3. Fold the slips, put them in the hat and mix.
  4. The reader draws one slip without looking.
  5. Add that word to the sheet. If the slip says STOP, the sentence is finished.
  6. Empty the hat and start the next turn.

Finish at STOP, or after 8 words.

Run it again

Start a new line with the same prompt and play again with the same rules and fresh slips every turn. Don’t plan ahead or try to change the result. Then compare your two sentences.

Keep duplicate words as separate slips: three “frog” slips give frog three chances. Never reuse old slips, because the sentence has changed. Two runs can match; that is a fair result.

Discussion

Which words appeared on more than one slip? Did one drawn word change what you wrote next? Why are your two sentences different? The slips in the hat were a probability distribution: every possible next word, with its chance of being drawn.

This models sampling during generation. People supply the proposals; a trained model calculates the chances from what it learned, and its tokens can be smaller than words. Allow approximately 12 minutes for setup, two runs and discussion. If time is tight, shorten the second run rather than changing the rules.

After each activity

What mechanism could learners explain before running it again? What could they change? What dependencies would remain if this became a real classroom tool?

No phone or account is required. A partner can dictate or read slips. The future phone-assisted version and microphone-triggered reveal are outside this draft.