## A small puzzle involving a Roberval balance

**Authors:** Jean-Christophe Filliâtre / Léon Gondelman

**Topics:** Ghost code

**Tools:** Why3

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# 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 import Roberval use import array.Array

All values in `balls(lo..hi-1)`

are equal to `w`

, apart from balls`lb`

which is smaller.

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

Solve the problem for 3 balls, using exactly one weighing.
The solution `lb`

is passed as a ghost argument.

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 8 balls, using exactly two weighings.
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 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 import Roberval use import array.Array

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.

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 end

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