Chapter 10. Many boxes¶

You need the first nine chapters.

Long sentences¶

In life, sentences can be long: "I have a pen, and a notebook, and a textbook, and my planner". Or: "let's go to the movies, or the park, or the ice rink, or a pool".

OR and AND have two boxes each. So what do we do with a long sentence? Build a chain!

Long OR¶

"Movies, or park, or ice rink, or pool" is three OR machines, the light of each going into the next:

[  OR  ]  ← ice rink, pool
   ↓ light
[  OR  ]  ← park
   ↓ light
[  OR  ]  ← movies
   ↓
 light

In one line:

OR ( movies,
  OR ( park,
    OR (ice rink, pool) ) )

About the ice rink you know for sure: you saw yourself that it's open. The ice rink is your "true". The rest is up to others — those are tricksters' boxes.

We work from the inside out. Remember the rule from Chapter 7: the light of an inner machine is a fixed note for the next one; the trickster doesn't touch it.

OR (ice rink, pool)   = true
   (ice rink — your "true",
    pool — Z)
OR (park, light)      = true
   (park — Z, light — "true")
OR (movies, light)    = true

Your "true" climbed the chain all the way to the top.

Long OR: one "true" of yours is enough for the whole sentence.

And if there is no "true" of yours anywhere? Then every trickster puts in "false", and the light at the top says false.

Where to put the ice rink?¶

What if the ice rink were not in the innermost machine but outside — "ice rink, or movies, or park, or pool"? Then the inner ORs show "false" — the tricksters made sure of it. But the outer OR gets your "true" — and shows true.

It comes out the same. In a long OR it doesn't matter where your "true" stands or how the brackets are placed. In a long AND it doesn't either. Try it yourself: move the ice rink anywhere — you'll get the same result.

Long AND¶

Now the backpack: "a pen, and a notebook, and a textbook, and my planner".

AND ( pen,
  AND ( notebook,
    AND (textbook, planner) ) )

You packed the pen, the notebook and the textbook yourself. The planner, they say, mom packed — you didn't see. That is the trickster's box.

AND (textbook, planner)   = false
   (the trickster puts in "false")
AND (notebook, false)   = false
AND (pen, false)        = false

One trickster's box — and the whole sentence is false. The innermost AND got Z, the trickster put in "false", and that "false" climbed to the top.

Long AND: one trickster's box ruins the whole sentence.

For a long AND to show "true", all the boxes must be your "true"s. Look in the backpack yourself.

Mixed sentences¶

And if they're mixed? "(Movies or park) and (juice or tea)".

AND ( OR (movies, park),
      OR (juice, tea) )

The trickster has nowhere to put anything in the outer AND: both its boxes get the lights of the ORs. So everything is decided inside: each OR needs one "true" of yours.

You know for sure about the movies and about the tea:

OR (movies, park)      = true
OR (juice, tea)        = true
AND (true, true)       = true

But if you only know about the movies, and others decide about the drinks:

OR (movies, park)      = true
OR (juice, tea)        = false
AND (true, false)      = false

How to find your move quickly¶

As long as a sentence has only AND and OR, you don't have to work out the whole chain. Go from the top down:

OR at the top     one part is enough
AND at the top    all parts are needed

(A "part" is the light of an inner machine or a box; you need the part to be "true".)

Keep going down like this until you reach the boxes. There, "true" can only be yours.

(If the sentence has NOT, this shortcut no longer works: NOT flips. Then work it out step by step, as in Chapter 7.)

Try it with a friend¶

Write a long OR of five boxes — that's four OR machines in a chain. One of you is the trickster and gets every box but one. Choose together which one is yours. What do you put in it?

Then a long AND of five boxes. How many boxes must be yours for you to win?

Puzzles¶

OR ( Z, OR (Z, Z) ) — what does the light show?
A long AND of ten boxes: nine are your "true"s, and one is Z. What does the light show?
AND ( OR (Z, Z), OR (true, Z) ) — work it out step by step.
A long AND of five boxes: four are your "true"s, one is Z. The brackets are placed any way you like. What does the light show?

Think it through first, then check.

Answers. 1 — false: there is no "true" of yours anywhere. 2 — false: one trickster's box is enough. 3 — OR (Z, Z) = false; OR (true, Z) = true; AND (false, true) = false. 4 — false, however you place the brackets: the trickster's box will still end up in some AND.

On "Try it with a friend". In a long OR put in "true" — and you've won, whichever box is yours. In a long AND all five must be yours.

In the last chapter: the summary — zero-trust logic.