Skip to content

Chapter 5. Where the trickster always wins

You need the first four chapters.

Is there a machine with no secret?

Each of our three machines had a secret. On OR put in "true", on NOT-AND "false", on IF… THEN it depends on which box is yours. Each time the trickster was left empty-handed.

Maybe every machine has a secret — you just have to look for it?

No. You are about to see machines on which the trickster always wins, whatever you put in. One box is yours, the other is his, and his box has the value Z.

The AND machine

Its light shows "true" only if both boxes hold "true":

first   second   light
true    true     true
true    false    false
false   true     false
false   false    false

You put in "true". The trickster puts in "false". The table: true, false — the light says false.

You put in "false". The trickster doesn't even need to think: with your "false" the light can no longer show "true". False.

In one line:

AND (true, Z)  = false
AND (false, Z) = false

The AND machine needs both boxes to agree. And the trickster never agrees.

The NOT-OR machine

Don't mix it up: this is not the OR machine with a NOT in front of it, but one machine, with its own table — like NOT-AND in Chapter 2. It shows "true" only if both boxes hold "false":

first   second   light
true    true     false
true    false    false
false   true     false
false   false    true

You put in "false". The trickster puts in "true" — false.

You put in "true". The light can no longer show "true" — false.

The EITHER-OR machine

It shows "true" when the notes are different. If they are the same — "false".

The trickster simply copies your note. You put in "true" — he puts in "true". You put in "false" — he puts in "false". The notes are the same — false.

The SAME machine

The opposite: "true" when the notes are the same.

The trickster puts in the opposite. The notes are different — false.

How to tell in advance

You don't need to play to find out whether a machine has a secret. The table is enough.

Put your finger on your note — say, "true" in the first box. Two rows of the table go with it: one for when the trickster puts in "true", the other for when he puts in "false". If the light says "true" in both rows, that is your secret. If even one of them says "false", the trickster will pick exactly that one.

Check both of your notes this way — "true" and "false". And if the machine's boxes are different, as with IF… THEN, check each box separately too.

Check the OR machine this way: with your "true" both rows say "true". Secret found. But with the AND machine, whatever your note, at least one row says "false". No secret.

There is a secret if your note guarantees "true" — whatever the trickster puts in.

How many machines are there?

Logicians have counted every two-box machine. There are only sixteen different tables: a table has four rows, each row's light is "true" or "false", and that makes 2 × 2 × 2 × 2 = 16.

But six of them are not real two-box machines:

always true        flip the first
always false       copy the second
copy the first     flip the second

The first two don't read the notes at all. The others read only one box: if it is yours, you decide everything; if it is his, he does.

That leaves ten real ones. Only four of them have a secret:

OR                     put in "true"
NOT-AND                put in "false"
IF… THEN               "if" — "false",
                       "then" — "true"
IF… THEN reversed      the same secret

"IF… THEN reversed" is the same machine, only with "if" on the right and "then" on the left. "If" and "then" play different roles, so these are two different tables — and they count as two. Four tables with a secret, but three machines.

On the other six the trickster always wins: AND, NOT-OR, EITHER-OR, SAME and two more that are like AND — "the first is true and the second is false" and the other way round.

That is why the first three chapters were about exactly these three machines.

An example from life

Remember from Chapter 1: "the test is tomorrow OR today is Monday"? The chat is the trickster's box, the calendar is yours. The calendar says "Monday" — true, and OR shows "true", whatever is posted in the chat.

Let's change the machine: "the test is tomorrow AND today is Monday". Same calendar, "true". But now the chat decides: if they post that there's no test, the whole sentence is false. ZTL answers: false.

That doesn't mean there definitely is no test. It means the trickster can make the sentence false. And in ZTL "true" happens only when he can't.

Try it with a friend

Take the AND machine. Let your friend be the trickster. Try putting in "true" and "false" as often as you like. Do you ever get "true"?

Then play EITHER-OR, swapping roles. The trickster copies his friend's note — see whether the friend ever gets "true".

Puzzles

  1. The AND machine. You put in "true". What does the trickster put in, and what does the light show?
  2. The EITHER-OR machine. You put in "false". What does the trickster put in?
  3. On which three machines can you stop the trickster?
  4. Write in one line: the SAME machine, your box holds "true", the other one is Z.
  5. The machine "the first is true and the second is false": its light shows "true" only in that case. Your box is the first one. Does it have a secret? Draw its table and check with your finger.

Think it through first, then check.

Answers. 1 — "false"; the light says "false". 2 — "false" too: the notes are the same, and the light says "false". 3 — OR, NOT-AND, IF… THEN. 4 — SAME (true, Z) = false. 5 — no: if you put in "true", the trickster puts "true" in the second box, and the light says "false"; if you put in "false", the light says "false" straight away.

On "Try it with a friend". On the AND machine — never: the trickster puts in "false". On EITHER-OR — never either: a copied note makes them the same.

In the next chapter: a machine with just one box — NOT.