## A small puzzle involving a Roberval balance

Topics: Ghost code

Tools: Why3

see also the index (by topic, by tool, by reference, by year)

# Two puzzles involving a Roberval balance

Note: Ghost code is used to get elegant specifications.

Jean-Christophe Filliâtre (CNRS), December 2013 Léon Gondelman (Université Paris-Sud), April 2014

```module Roberval

use export int.Int
use export ref.Refint

type outcome = Left | Equal | Right
```

the side of the heaviest mass i.e. where the balance leans

```  val ghost counter: ref int
```

how many times can we use the balance

```  val balance (left right: int) : outcome
requires { !counter > 0 }
ensures  { match result with
| Left  -> left > right
| Equal -> left = right
| Right -> left < right
end }
writes   { counter }
ensures  { !counter = old !counter - 1 }

end

```

You are given 8 balls and a Roberval balance. All balls have the same weight, apart from one, which is lighter. Using the balance at most twice, determine the lighter ball.

Though this problem is not that difficult (though, you may want to think about it before reading any further), it is an interesting exercise in program specification.

The index of the lighter ball is passed as a ghost argument to the program. Thus it cannot be used to compute the answer, but only to write the specification.

```module Puzzle8

use Roberval
use array.Array

predicate spec (balls: array int) (lo hi: int) (lb w: int) =
0 <= lo <= lb < hi <= length balls &&
(forall i: int. lo <= i < hi -> i <> lb -> balls[i] = w) &&
balls[lb] < w
```

All values in `balls[lo..hi-1]` are equal to `w`, apart from `balls[lb]` which is smaller.

```  let solve3 (balls: array int) (lo: int) (ghost lb: int) (ghost w: int) : int
requires { !counter >= 1 }
requires { spec balls lo (lo + 3) lb w }
ensures  { result = lb }
ensures  { !counter = old !counter - 1 }
=
match balance balls[lo] balls[lo+1] with
| Right -> lo
| Left  -> lo+1
| Equal -> lo+2
end
```

Solve the problem for 3 balls, using exactly one weighing. The solution `lb` is passed as a ghost argument.

```  let solve8 (balls: array int) (ghost lb: int) (ghost w: int) : int
requires { !counter = 2 }
requires { spec balls 0 8 lb w }
ensures  { result = lb }
=
(* first, compare balls 0,1,2 with balls 3,4,5 *)
match balance (balls[0] + balls[1] + balls[2])
(balls[3] + balls[4] + balls[5]) with
| Right -> solve3 balls 0 lb w
| Left  -> solve3 balls 3 lb w
(* 0,1,2 = 3,4,5 thus lb must be 6 or 7 *)
| Equal -> match balance balls[6] balls[7] with Right -> 6 | _ -> 7 end
end
```

Solve the problem for 8 balls, using exactly two weighings. The solution `lb` is passed as a ghost argument.

```end

```

You are given 12 balls, all of the same weight except one (for which you don't knwo whether it is lighter or heavier)

Given a Roberval balance, you have to find the intruder, and determine whether it is lighter or heavier, using the balance at most three times.

```module Puzzle12

use Roberval
use array.Array

let solve12 (balls: array int) (ghost w j: int) (ghost b: bool) : (int, bool)
requires { !counter = 3 }
requires { 0 <= j < 12 = length balls }
requires { forall i: int. 0 <= i < 12 -> i <> j -> balls[i] = w }
requires { if b then balls[j] < w else balls[j] > w }
ensures  { result = (j, b) }
=
match balance (balls[0] + balls[1] + balls[2] + balls[3])
(balls[4] + balls[5] + balls[6] + balls[7]) with
| Equal -> (* 0,1,2,3 = 4,5,6,7 *)
match balance (balls[0] + balls[8]) (balls[9] + balls[10]) with
| Equal -> (* 0,8 = 9,10 *)
match balance balls[0] balls[11] with
| Right -> 11, False | _ -> 11, True end
| Right -> (* 0,8 < 9,10 *)
match balance balls[9] balls[10] with
| Equal -> 8, True
| Right -> 10,  False
| Left  -> 9, False
end
| Left -> (* 0,8 > 9,10 *)
match balance balls[9] balls[10] with
| Equal -> 8, False
| Right -> 9,  True
| Left  -> 10, True
end
end
| Right -> (* 0,1,2,3 < 4,5,6,7 *)
match balance (balls[0] + balls[1] + balls[4])
(balls[2] + balls[5] + balls[8]) with
| Equal -> (* 0,1,4 = 2,5,8 *)
match balance balls[6] balls[7] with
| Equal -> 3, True
| Right -> 7, False
| Left -> 6, False
end
| Right -> (* 0,1,4 < 2,5,8 *)
match balance balls[0] balls[1] with
| Equal -> 5, False
| Right -> 0, True
| Left -> 1, True
end
| Left -> (* 0,1,4 > 2,5,8 *)
match balance balls[4] balls[8] with
| Equal -> 2, True
| _     -> 4, False
end
end
| Left -> (* 0,1,2,3 > 4,5,6,7 *)
match balance (balls[0] + balls[1] + balls[4])
(balls[2] + balls[5] + balls[8]) with
| Equal -> (* 0,1,4 = 2,5,8 *)
match balance balls[6] balls[7] with
| Equal -> 3, False
| Right -> 6, True
| Left -> 7, True
end
| Right -> (* 0,1,4 < 2,5,8 *)
match balance balls[2] balls[5] with
| Equal -> 4, True
| Right -> 5, False
| Left -> 2, False
end
| Left -> (* 0,1,4 > 2,5,8 *)
match balance balls[0] balls[1] with
| Equal -> 5, True
| Right -> 1, False
| Left -> 0, False
end
end
end
```

The index `j` of the intruder, its status `b` (whether it is lighter or heavier), and the weight `w` of the other balls are all passed as ghost arguments.

```end
```