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    <p>Here are solutions for each starting position:</p> <pre><code>Start 1: (2, 0, -1) Reds=1 Blues=0 Red B ==&gt; (3, -1, -2) Reds=1 Blues=0 Red B ==&gt; (4, -2, -3) Reds=1 Blues=0 Red D ==&gt; (4, 0, -2) Reds=0 Blues=2 Blue A ==&gt; (2, 0, -1) Reds=0 Blues=1 Blue A ==&gt; (0, 0, 0) Reds=0 Blues=0 Start 2: (3, 1, -1) Reds=2 Blues=0 Red A ==&gt; (3, -1, -2) Reds=1 Blues=0 Red B ==&gt; (4, -2, -3) Reds=1 Blues=0 Red D ==&gt; (4, 0, -2) Reds=0 Blues=2 Blue A ==&gt; (2, 0, -1) Reds=0 Blues=1 Blue A ==&gt; (0, 0, 0) Reds=0 Blues=0 </code></pre> <p>Found instantly by the following C# program that does a random walk and gives up after a small number of moves.</p> <pre><code>using System; using System.Collections.Generic; using System.Linq; using System.Text; namespace SO8683939 { struct State { public int V1; public int V2; public int V3; public int Reds; public int Blues; public int Tokens { get { return Reds + Blues; } } public string Description; public State(int v1, int v2, int v3, int reds, int blues) { V1 = v1; V2 = v2; V3 = v3; Reds = reds; Blues = blues; Description = null; } public State Add(State other) { State sum; sum.V1 = V1 + other.V1; sum.V2 = V2 + other.V2; sum.V3 = V3 + other.V3; sum.Reds = Reds + other.Reds; sum.Blues = Blues + other.Blues; sum.Description = null; return sum; } public override string ToString() { var detail = string.Format("({0}, {1}, {2}) Reds={3} Blues={4}", V1, V2, V3, Reds, Blues); if (Description != null) { return Description + ": " + detail; } return detail; } } class Program { static void Main(string[] args) { var start1 = new State(2, 0, -1, 1, 0) { Description = "Start 1" }; var start2 = new State(3, 1, -1, 2, 0) { Description = "Start 2" }; var end = new State(0, 0, 0, 0, 0); var redA = new State(0, -2, -1, -1, 0) { Description = "Red A" }; var redB = new State(1, -1, -1, 0, 0) { Description = "Red B" }; ; var redC = new State(2, 0, -1, 1, 0) { Description = "Red C" }; ; var redD = new State(0, 2, 1, -1, 2) { Description = "Red D" }; ; var redOptions = new[] { redA, redB, redC, redD }; var blueA = new State(-2, 0, 1, 0, -1) { Description = "Blue A" }; var blueB = new State(-1, 1, 1, 0, 0) { Description = "Blue B" }; var blueC = new State(0, 2, 1, 0, 1) { Description = "Blue C" }; var blueD = new State(2, 0, -1, 2, -1) { Description = "Blue D" }; var blueOptions = new[] { blueA, blueB, blueC, blueD }; var startingPosition = start1; var maxSolutionLength = 5; var rand = new Random(); var path = new List&lt;State&gt;(); while (true) { var current = startingPosition; path.Clear(); //Console.WriteLine("Starting"); //Console.WriteLine(current); while (true) { State selected; if (current.Reds == 0) { selected = blueOptions[rand.Next(4)]; } else if (current.Blues == 0) { selected = redOptions[rand.Next(4)]; } else { if (rand.NextDouble() &lt; 0.5) { selected = blueOptions[rand.Next(4)]; } else { selected = redOptions[rand.Next(4)]; } } //Console.WriteLine(selected); path.Add(selected); current = current.Add(selected); //Console.WriteLine(current); if (current.Equals(end)) { Console.WriteLine("Success!"); var retrace = startingPosition; Console.WriteLine(retrace); foreach (var selection in path) { retrace = retrace.Add(selection); Console.WriteLine("{0} ==&gt; {1}", selection.Description, retrace); } Console.ReadLine(); break; } else if (current.Tokens == 0) { // fail //Console.WriteLine("Fail"); break; } else if (path.Count &gt;= maxSolutionLength) { // fail //Console.WriteLine("Fail"); break; } } } } } } </code></pre>
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