Countdown

Countdown is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Task

Given six numbers randomly selected from the list [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 25, 50, 75, 100], calculate using only positive integers and four operations [+, -, *, /] a random number between 101 and 999.

Example:

Using: [3, 6, 25, 50, 75, 100]
Target: 952

Solution:

• 100 + 6 = 106
• 75 * 3 = 225
• 106 * 225 = 23850
• 23850 - 50 = 23800
• 23800 / 25 = 952

Origins

This is originally a 1972 French television game show. The game consists of randomly selecting six of the twenty-four numbers, from a list of: twenty "small numbers" (two each from 1 to 10), and four "large numbers" of 25, 50, 75 and 100. A random target number between 101 and 999 is generated. The players have 30 seconds to work out a sequence of calculations with the numbers whose final result is as close as possible to the target number. Only the four basic operations: addition, subtraction, multiplication and division can be used to create new numbers and not all six numbers are required. A number can only be used once. Division can only be done if the result has no remainder (fractions are not allowed) and only positive integers can be obtained at any stage of the calculation. (More info on the original game).

Extra challenge

The brute force algorithm is quite obvious. What is more interesting is to find some optimisation heuristics to reduce the number of calculations. For example, a rather interesting computational challenge is to calculate, as fast as possible, all existing solutions (that means 2'764'800 operations) for all possible games (with all the 13'243 combinations of six numbers out of twenty-four for all 898 possible targets between 101 and 999).

Quorum

use Libraries.Containers.List
use Libraries.Containers.Iterator
use Libraries.System.DateTime

action Main
    DateTime datetime
    number start = datetime:GetEpochTime() 
    List<integer> numbers
    numbers:Add(100)
    numbers:Add(75)
    numbers:Add(50)
    numbers:Add(25)
    numbers:Add(6)
    numbers:Add(3)
    Solution(952,numbers)
    number stop = datetime:GetEpochTime() 
    output stop-start + " ms"
end

action Solution(integer target, List<integer> numbers) returns boolean
    
    if numbers:GetSize() = 1
        return false
    end    

    Iterator<integer> it0 = numbers:GetIterator()
    repeat while it0:HasNext()

        integer n0 = it0:Next()
        List<integer> numbers1 = cast(List<integer>, numbers:Copy())   
        numbers1:Remove(n0)

        Iterator<integer> it1 = numbers1:GetIterator()
        repeat while it1:HasNext()

            integer n1 = it1:Next()
            List<integer> numbers2 = cast(List<integer>, numbers1:Copy())
            numbers2:Remove(n1)

            integer res = 0
            List<integer> numbersNew

            res = n0 + n1
            numbersNew = cast(List<integer>, numbers2:Copy())
            numbersNew:Add(res)
            if res = target or Solution(target, numbersNew)
                output res + " = " + n0 + " + " + n1
                return true
            end

            res = n0 * n1
            numbersNew = cast(List<integer>, numbers2:Copy())
            numbersNew:Add(res)
            if res = target or Solution(target, numbersNew)
                output res + " = " + n0 + " * " + n1
                return true
            end

            if n0 < n1
                integer temp = n0
                n0 = n1
                n1 = temp
            end

            if n0 not= n1
                res = n0 - n1
                numbersNew = cast(List<integer>, numbers2:Copy())
                numbersNew:Add(res)
                if res = target or Solution(target, numbersNew)
                    output res + " = " + n0 + " - " + n1
                    return true
                end
            end

            if n0 mod n1 = 0
                res = n0 / n1
                numbersNew = cast(List<integer>, numbers2:Copy())
                numbersNew:Add(res)
                if res = target or Solution(target, numbersNew)
                    output res + " = " + n0 + " / " + n1
                    return true
                end
            end
        end
    end
    return false
end

Output:

952 = 50 + 902
902 = 22550 / 25
22550 = 50 + 22500
22500 = 7500 * 3
7500 = 100 * 75
214.0 ms

Wren

Translation of: Quorum

Library: Wren-fmt

This is based on the original Quorum algorithm as it's more or less the approach I'd have used anyway.

The latest algorithm is not working properly as numbers are being used twice (50 in the first example).

A bit slow but not too bad for Wren :)

import "random" for Random
import "./fmt" for Fmt

var countdown // recursive function
countdown = Fn.new { |numbers, target|
    if (numbers.count == 1) return false
    for (n0 in numbers) {
        var nums1 = numbers.toList
        nums1.remove(n0)
        for (n1 in nums1) {
            var nums2 = nums1.toList
            nums2.remove(n1)
            var res = n0 + n1
            var numsNew = nums2.toList
            numsNew.add(res)
            if (res == target || countdown.call(numsNew, target)) {
                Fmt.print("$d = $d + $d", res, n0, n1)
                return true
            }

            res = n0 * n1
            numsNew = nums2.toList
            numsNew.add(res)
            if (res == target || countdown.call(numsNew, target)) {
                Fmt.print("$d = $d * $d", res, n0, n1)
                return true
            }

            if (n0 > n1) {
                res = n0 - n1
                numsNew = nums2.toList
                numsNew.add(res)
                if (res == target || countdown.call(numsNew, target)) {
                    Fmt.print("$d = $d - $d", res, n0, n1)
                    return true
                }
            } else if (n1 > n0) {
                res = n1 - n0
                numsNew = nums2.toList
                numsNew.add(res)
                if (res == target || countdown.call(numsNew, target)) {
                    Fmt.print("$d = $d - $d", res, n1, n0)
                    return true
                }
            }

            if (n0 > n1) {
                if (n0 % n1 == 0) {
                    res = n0 / n1
                    numsNew = nums2.toList
                    numsNew.add(res)
                    if (res == target || countdown.call(numsNew, target)) {
                        Fmt.print("$d = $d / $d", res, n0, n1)
                        return true
                    }
                }
            } else {
                if (n1 % n0 == 0) {
                    res = n1 / n0
                    numsNew = nums2.toList
                    numsNew.add(res)
                    if (res == target || countdown.call(numsNew, target)) {
                        Fmt.print("$d = $d / $d", res, n1, n0)
                        return true
                    }
                }
            }
        }
    }
    return false
}

var allNumbers = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 25, 50, 75, 100]
var rand = Random.new()
var numbersList = [
    [3, 6, 25, 50, 75, 100],
    [100, 75, 50, 25, 6, 3], // see if there's much difference if we reverse the first example
    [8, 4, 4, 6, 8, 9],
    rand.sample(allNumbers, 6)
]
var targetList = [952, 952, 594, rand.int(101, 1000)]
for (i in 0...numbersList.count) {
    System.print("Using : %(numbersList[i])")
    System.print("Target: %(targetList[i])")
    var start = System.clock
    var done = countdown.call(numbersList[i], targetList[i])
    System.print("Took %(((System.clock - start) * 1000).round) ms")
    if (!done) System.print("No solution exists")
    System.print()
}

Output:

Sample output (as the fourth example is random):

Using : [3, 6, 25, 50, 75, 100]
Target: 952
952 = 23800 / 25
23800 = 23850 - 50
23850 = 225 * 106
106 = 6 + 100
225 = 3 * 75
Took 1525 ms

Using : [100, 75, 50, 25, 6, 3]
Target: 952
952 = 23800 / 25
23800 = 23850 - 50
23850 = 106 * 225
225 = 75 * 3
106 = 100 + 6
Took 1522 ms

Using : [8, 4, 4, 6, 8, 9]
Target: 594
594 = 54 * 11
11 = 8 + 3
54 = 6 * 9
3 = 12 / 4
12 = 8 + 4
Took 27 ms

Using : [2, 4, 9, 10, 3, 5]
Target: 363
363 = 3 + 360
360 = 9 * 40
40 = 10 + 30
30 = 5 * 6
6 = 2 + 4
Took 107 ms