Optimum Letter Cards for Spelling Words Let's say you have a list of words and you want to be able to use letter cards to spell each word. For example, to spell cat, you would use three cards labelled C, A, T.

Assuming each card is double-sided, submit a program to define a minimum number of cards that can be used to spell the entire list of words.

Input is the word list, it can be file-based, hard-coded, command line, whatever. Output is the list of cards, formatted and ordered as you see fit, provided it is clear how the cards are labelled.

Case is not significant: Golf, golf and GOLF are equivalent.

Some hints:


the number of cards can be no less than the length of the longest word
it makes no sense for a card to have the same letter on both sides
while case is not significant, recommend lower case to take advantage of certain symmetries


Examples, these take advantage of certain symmetries:

Input: ben, bog, bug, den, do, doe, dog, due, dug, Ed, end, gob, God, Ned, ode, pen, Poe, pug

Output: b/d, e/g, o/n

Input: an, and, ape, are, be, bed, bud, bur, Dan, Deb, dub, ear, Ed, era, nap, pan, pea, pub, Rae, ran, rub

Output: a/b, d/r, e/n

Making it a popularity contest, so elegance of code, run-time performance, and cleverness (including rule-bending &amp; loopholes) are important!

Addition: Some have asked about "allowed" symmetries, whether special fonts can be used, and whether the cards can be folded.

Allowed symmetries are any letters which look similar to each other after 0, 90, 180 or 270 degrees rotation. This includes b/q, d/p and n/u. I would also say M/W, Z/N, and of course I/l (capital i, lowercase L). I'm probably scratching the surface, so if there are any others you're unsure about, just ask.

To keep it simple, please restrict to a standard sans-serif font, say that used in SE.

As far as folding, while you can do some amazing substitutions, e.g. B can be D, E, F, I, P, or R, and maybe C or L if you fold really creatively, I think that's bending, literally, too much!

I came up with this problem while playing with some similar cards with my kids. I noted how easy it was to come up with single-sided cards vs. how difficult it was to come up with double-sided cards.

Addition: Have provided a bounty to be awarded to the most popular answer. If there's a tie, will award to the one who submitted first.

Another hint:


solving the single-sided problem will give you an idea of the minimum number of cards needed (e.g. 20 single-sided cards translates to at least 10 double sided-cards needed)


Addition: Oh bother, I was busy and forgot about the bounty expiring. It ended up going to no one because the only answer was submitted before the bounty started! Sorry about that.
C# - CardChooser

Summary

This application uses a brute force method to attempt to solve each list. First I create a list of potential cards to select from, then I determine which is a best fit(removes the most chars + shortens long words most), add this to a result list and continue with this process until I have selected enough potential cards to remove every word in the list, then I  rematch those cards to each word and print the output.

If you would like to see a more limited version of this code without downloading and building the provided windows forms application you can use the link provided to run my program on smaller sets of data, please note that this is the console application version so the resulting cards are NOT rotated:
http://ideone.com/fork/VD1gJF

Revision History

Current - Added better result optimization suggested by @Zgarb

Update 3 - More code cleanup, more bugs fixed, better results

Update 2 - Windows Forms, More verbose output

Update 1 - New/Better support for character symmetries

Original - Console Application

Examples

acr, aft, ain, sll, win, say, said, fast, epic


hes, will, with, wont, would, wouldve, wouldnt, yet, you, youd, youll


aaaa, bbbb, cccc

Code

I still need to combine this into one larger project with the ConsoleApp and WindowsForms code all sharing the same classes and methods, then split out the different regions in the RunButton_Click method so I can write units around them, anyway whenever I find time to do that I will, for now this is what I have: 

using System;
using System.Collections.Generic;
using System.ComponentModel;
using System.Data;
using System.Drawing;
using System.Drawing.Imaging;
using System.IO;
using System.Linq;
using System.Text;
using System.Windows.Forms;

namespace CardChooserForms
{
    public partial class CardChooser : Form
    {
        private class Solution : IEquatable&lt;Solution&gt;
        {
            public List&lt;string&gt; Cards { get; set; }
            public List&lt;string&gt; Remaining { get; set; }

            public int RemainingScore
            {
                get
                {
                    return this.Remaining.Sum(b =&gt; b.ToCharArray().Count());
                }
            }

            public bool Equals(Solution other)
            {
                return new string(Cards.OrderBy(a =&gt; a).SelectMany(a =&gt; a).ToArray()) == new string(other.Cards.OrderBy(a =&gt; a).SelectMany(a =&gt; a).ToArray());
            }

            public override int GetHashCode()
            {
                return (new string(Cards.OrderBy(a =&gt; a).SelectMany(a =&gt; a).ToArray())).GetHashCode();
            }
        }
        private class Symmetry
        {
            public char Value { get; set; }
            public Int16 RotationDifference { get; set; }
        }

        /// &lt;summary&gt;
        /// This is where Symmetries are stored, right now it only has support for pairs(two values per array)
        /// &lt;/summary&gt;
        private static Symmetry[][] _rotatableCharacters = new Symmetry[][] {                 
                new Symmetry[] { new Symmetry {Value = 'Z'}, new Symmetry {Value = 'N', RotationDifference = 90}}, 
                new Symmetry[] { new Symmetry {Value = 'd'}, new Symmetry {Value = 'p', RotationDifference = 180 }}, 
                new Symmetry[] { new Symmetry {Value = 'u'}, new Symmetry {Value = 'n', RotationDifference = 180 }}, 
                new Symmetry[] { new Symmetry {Value = 'm'}, new Symmetry {Value = 'w', RotationDifference = 180 }}, 
                new Symmetry[] { new Symmetry {Value = 'b'}, new Symmetry {Value = 'q', RotationDifference = 180 }}, 
                new Symmetry[] { new Symmetry {Value = 'l'}, new Symmetry {Value = 'I', RotationDifference = 0}},                 
            };

        //These all control the output settings
        private readonly static int _defualtSpacing = 25;
        private readonly static int _defualtFontSize = 8;
        private readonly static Font _defualtFont = new Font("Microsoft Sans Serif", _defualtFontSize);
        private readonly static Brush _defualtBackgroundColor = Brushes.Beige;
        private readonly static Brush _defualtForegroundColor = Brushes.Black;


        public CardChooser()
        {
            InitializeComponent();
        }

        private void RunButton_Click(object sender, EventArgs e)
        {            
            #region Input Parsing
            //Get input                         
            string input = InputRichTextBox.Text;

            if (!input.Contains(","))
                throw new ArgumentException("Input must contain more than one value and must be seprated by commas.");

            //Parse input
            var inputLowercasedTrimedTransformed = input.Split(',').Select(a =&gt; a.ToLowerInvariant().Trim()).ToArray();
            var inputSplitTrimIndex = input.Split(',').Select(a =&gt; a.Trim()).ToArray().Select((a, index) =&gt; new { value = a, index }).ToArray();
            #endregion Input Parsing

            #region Card Formation
            var inputCharParsed = inputLowercasedTrimedTransformed.Select(a =&gt; a.ToCharArray()).ToArray();
            var possibleCards = GetAllCasesTwoLengthArrayElements(
                UniqueBiDirection(
                //Get unique characters
                    inputCharParsed
                    .SelectMany(a =&gt; a)
                    .Distinct()
                    .Select(a =&gt; new
                    {
                        Character = a,
                        PossibleCharacters = inputCharParsed.SelectMany(b =&gt; b).Where(b =&gt; b != a).ToList()
                    })
                //Now get distinct cards(ie NB == BN, NB != NE)
                    .SelectMany(a =&gt; a.PossibleCharacters.Select(b =&gt; new string(new char[] { a.Character, b })).ToArray()).ToArray()
                    ).ToArray()
                ).ToArray();

            //Now get every possible character each card can eliminate
            var possibleCharsFromCards = GetAllPossibleCharsFromACards(possibleCards).ToArray();

            //Now set up some possibilities that contain only one card
            var possibleCardCombinations = possibleCards.Select((a, index) =&gt; new Solution
            {
                Cards = new List&lt;string&gt; { a },
                //Use the index of each card to reference the possible characters it can remove, then remove them per card to form a initial list of cards
                Remaining = inputLowercasedTrimedTransformed.Select(b =&gt; b.RemoveFirstInCharArr(possibleCharsFromCards[index].ToLowerInvariant().ToCharArray())).ToList()
            })
            //Take the best scoring card, discard the rest
            .OrderBy(a =&gt; a.RemainingScore)
            .ThenBy(a =&gt; a.Remaining.Max(b =&gt; b.Length))
            .Take(1).ToList();
            #endregion Card Formation

            #region Card Selection
            //Find best combination by iteratively trying every combination + 1 more card, and choose the lowest scoring one 
            while (!possibleCardCombinations.Any(a =&gt; a.Remaining.Sum(b =&gt; b.ToCharArray().Count()) == 0) &amp;&amp; possibleCardCombinations.First().Cards.Count() &lt; possibleCards.Count())
            {
                //Clear the list each iteration(as you can assume the last generations didn't work
                var newPossibilites = new List&lt;Solution&gt;();
                var currentRoundCardCombinations = possibleCardCombinations.ToArray();
                possibleCardCombinations.Clear();

                foreach (var trySolution in currentRoundCardCombinations)
                    foreach (var card in possibleCards.Select((a, index) =&gt; new { value = a, index }).Where(a =&gt; !trySolution.Cards.Contains(a.value)).ToArray())
                    {
                        var newSolution = new Solution();
                        newSolution.Cards = trySolution.Cards.ToList();
                        newSolution.Cards.Add(card.value);
                        newSolution.Remaining = trySolution.Remaining.ToList().Select(a =&gt; a.RemoveFirstInCharArr(possibleCharsFromCards[card.index].ToLowerInvariant().ToCharArray())).ToList();
                        newPossibilites.Add(newSolution);
                    }

                //Choose the highest scoring card
                possibleCardCombinations = newPossibilites
                    .OrderBy(a =&gt; a.RemainingScore)
                    .ThenBy(a =&gt; a.Remaining.Max(b =&gt; b.Length))
                    .Distinct().Take(1).ToList();
            }
            var finalCardSet = possibleCardCombinations.First().Cards.ToArray();
            #endregion Card Selection

            #region Output
            using (var image = new Bitmap(500, inputSplitTrimIndex.Count() * _defualtSpacing + finalCardSet.Count() * (_defualtFontSize / 2) + _defualtSpacing))
            using (Graphics graphic = Graphics.FromImage(image))
            {
                //Background
                graphic.FillRectangle(_defualtBackgroundColor, 0, 0, image.Width, image.Height);

                //Header                
                graphic.DrawString("Total Number of Cards Required: " + finalCardSet.Count(), _defualtFont, _defualtForegroundColor, new PointF(0, 0));
                graphic.DrawString(
                    "Cards: " + String.Join(", ", finalCardSet.Select(a =&gt; a[0] + "/" + a[1])),
                    _defualtFont,
                    _defualtForegroundColor,
                    new RectangleF(0, _defualtSpacing, image.Width - _defualtSpacing, finalCardSet.Count() * 5));

                //Results
                foreach (var element in inputSplitTrimIndex)
                {
                    //Paint the word
                    graphic.DrawString(element.value + " -&gt; ", _defualtFont, _defualtForegroundColor, new PointF(0, element.index * _defualtSpacing + finalCardSet.Count() * (_defualtFontSize / 2) + _defualtSpacing));

                    //Now go through each character, determining the matching card, and wether that card has to be flipped
                    foreach (var card in GetOrderedCardsRequired(inputLowercasedTrimedTransformed[element.index].ToLowerInvariant(), finalCardSet.ToArray()).ToArray().Select((a, index) =&gt; new { value = a, index }))
                        using (var tempGraphic = Graphics.FromImage(image))
                        {
                            //For cards that need to flip
                            if (Char.ToUpperInvariant(element.value[card.index]) != Char.ToUpperInvariant(card.value[0]) &amp;&amp;
                                Char.ToUpperInvariant(element.value[card.index]) != Char.ToUpperInvariant(card.value[1]))
                            {
                                //TODO this is hacky and needs to be rethought
                                var rotateAmount = _rotatableCharacters
                                    .OrderByDescending(a =&gt; a.Any(b =&gt; b.Value == Char.ToLowerInvariant(element.value[card.index])))
                                    .First(a =&gt; a.Any(b =&gt; Char.ToUpperInvariant(b.Value) == Char.ToUpperInvariant(element.value[card.index])))
                                    [1].RotationDifference;

                                //Rotate
                                tempGraphic.TranslateTransform(
                                    _defualtSpacing * (_defualtFontSize / 2) + card.index * _defualtSpacing + (rotateAmount == 90 ? 0 : _defualtSpacing / 2) + (rotateAmount == 180 ? -(_defualtSpacing / 4) : 0),
                                    finalCardSet.Count() * (_defualtFontSize / 2) + _defualtSpacing + element.index * _defualtSpacing + (rotateAmount == 180 ? 0 : _defualtSpacing / 2));
                                tempGraphic.RotateTransform(rotateAmount);

                                //Print string
                                tempGraphic.DrawString(
                                String.Join("/", card.value.ToCharArray().Select(a =&gt; new string(new char[] { a })).ToArray()),
                                _defualtFont,
                                Brushes.Black,
                                new RectangleF(-(_defualtSpacing / 2), -(_defualtSpacing / 2), _defualtSpacing, _defualtSpacing));
                            }
                            else
                                tempGraphic.DrawString(
                                     String.Join("/", card.value.ToCharArray().Select(a =&gt; new string(new char[] { a })).ToArray()),
                                     _defualtFont,
                                     _defualtForegroundColor,
                                     new RectangleF(
                                         _defualtSpacing * (_defualtFontSize / 2) + card.index * _defualtSpacing,
                                         finalCardSet.Count() * (_defualtFontSize / 2) + _defualtSpacing + element.index * _defualtSpacing,
                                         _defualtSpacing, _defualtSpacing));
                        }
                }

                OutputPictureBox.Image = new Bitmap(image);
            }
            #endregion Output
        }

        private IEnumerable&lt;string&gt; GetAllPossibleCharsFromACards(string[] cards)
        {
            return cards.Select(a =&gt; 
                new string(a.ToCharArray().Concat(_rotatableCharacters
                                    .Where(b =&gt; b.Select(c =&gt; c.Value).Intersect(a.ToCharArray()).Count() &gt; 0)
                                    .SelectMany(b =&gt; b.Select(c =&gt; c.Value))
                                    .Distinct().ToArray()).Distinct().ToArray()));
        }

        private IEnumerable&lt;string&gt; GetOrderedCardsRequired(string word, string[] cards)
        {
            var solution = new List&lt;string&gt;();
            var tempCards = GetAllPossibleCharsFromACards(cards).Select((a, index) =&gt; new { value = a, index }).ToList();

            foreach (var letter in word.ToCharArray())
            {
                //TODO this still could theoretically fail I think                
                var card = tempCards
                    //Order by the least number of characters match
                    .OrderBy(a =&gt; word.ToLowerInvariant().Intersect(a.value.ToLowerInvariant()).Count())
                    .ThenByDescending(a =&gt; tempCards.Sum(b =&gt; b.value.ToLowerInvariant().Intersect(a.value.ToLowerInvariant()).Count()))
                    //Then take the least useful card for the other parts of the word
                    .First(a =&gt; a.value.ToLowerInvariant().Contains(Char.ToLowerInvariant(letter)));
                solution.Add(cards[card.index]);
                tempCards.Remove(card);
            }
            return solution;
        }

        private static IEnumerable&lt;string&gt; UniqueBiDirection(string[] input)
        {
            var results = new List&lt;string&gt;();
            foreach (var element in input)
                if (!results.Any(a =&gt; a == new string(element.ToCharArray().Reverse().ToArray()) || a == element))
                    results.Add(element);
            return results;
        }

        private static IEnumerable&lt;string&gt; GetAllCasesTwoLengthArrayElements(string[] input)
        {
            if (input.Any(a =&gt; a.Length != 2))
                throw new ArgumentException("This method is only for arrays with two characters");

            List&lt;string&gt; output = input.ToList();
            foreach (var element in input)
            {
                output.Add(new string(new char[] { Char.ToUpperInvariant(element[0]), Char.ToUpperInvariant(element[1]) }));
                output.Add(new string(new char[] { element[0], Char.ToUpperInvariant(element[1]) }));
                output.Add(new string(new char[] { Char.ToUpperInvariant(element[0]), element[1] }));
            }
            return output;
        }

        private void SaveButton_Click(object sender, EventArgs e)
        {
            using (var image = new Bitmap(OutputPictureBox.Image))
                image.Save(Directory.GetCurrentDirectory() + "Output.png", ImageFormat.Png);
        }
    }

    public static class StringExtensions
    {
        public static string RemoveFirstInCharArr(this string source, char[] values)
        {
            var tempSource = source.ToUpperInvariant();
            foreach (var value in values)
            {
                int index = tempSource.IndexOf(Char.ToUpperInvariant(value));
                if (index &gt;= 0) return source.Remove(index, 1);
            }
            return source;
        }        
    }
}

using System;
using System.Collections.Generic;
using System.Linq;
using System.Threading.Tasks;
using System.Windows.Forms;

namespace CardChooserForms
{
    static class Program
    {
        /// &lt;summary&gt;
        /// The main entry point for the application.
        /// &lt;/summary&gt;
        [STAThread]
        static void Main()
        {
            Application.EnableVisualStyles();
            Application.SetCompatibleTextRenderingDefault(false);
            Application.Run(new CardChooser());
        }
    }
}


namespace CardChooserForms
{
    partial class CardChooser
    {
        /// &lt;summary&gt;
        /// Required designer variable.
        /// &lt;/summary&gt;
        private System.ComponentModel.IContainer components = null;

        /// &lt;summary&gt;
        /// Clean up any resources being used.
        /// &lt;/summary&gt;
        /// &lt;param name="disposing"&gt;true if managed resources should be disposed; otherwise, false.&lt;/param&gt;
        protected override void Dispose(bool disposing)
        {
            if (disposing &amp;&amp; (components != null))
            {
                components.Dispose();
            }
            base.Dispose(disposing);
        }

        #region Windows Form Designer generated code

        /// &lt;summary&gt;
        /// Required method for Designer support - do not modify
        /// the contents of this method with the code editor.
        /// &lt;/summary&gt;
        private void InitializeComponent()
        {
            this.InputRichTextBox = new System.Windows.Forms.RichTextBox();
            this.EnterInputLabel = new System.Windows.Forms.Label();
            this.RunButton = new System.Windows.Forms.Button();
            this.OutputPictureBox = new System.Windows.Forms.PictureBox();
            this.OutputPanel = new System.Windows.Forms.Panel();
            this.SaveButton = new System.Windows.Forms.Button();
            ((System.ComponentModel.ISupportInitialize)(this.OutputPictureBox)).BeginInit();
            this.OutputPanel.SuspendLayout();
            this.SuspendLayout();
            // 
            // InputRichTextBox
            // 
            this.InputRichTextBox.Location = new System.Drawing.Point(60, 40);
            this.InputRichTextBox.Name = "InputRichTextBox";
            this.InputRichTextBox.Size = new System.Drawing.Size(400, 100);
            this.InputRichTextBox.TabIndex = 0;
            this.InputRichTextBox.Text = "";
            // 
            // EnterInputLabel
            // 
            this.EnterInputLabel.AutoSize = true;
            this.EnterInputLabel.Font = new System.Drawing.Font("Microsoft Sans Serif", 10F, System.Drawing.FontStyle.Regular, System.Drawing.GraphicsUnit.Point, ((byte)(0)));
            this.EnterInputLabel.Location = new System.Drawing.Point(57, 20);
            this.EnterInputLabel.Name = "EnterInputLabel";
            this.EnterInputLabel.Size = new System.Drawing.Size(81, 17);
            this.EnterInputLabel.TabIndex = 1;
            this.EnterInputLabel.Text = "Enter Input:";
            // 
            // RunButton
            // 
            this.RunButton.Font = new System.Drawing.Font("Microsoft Sans Serif", 20F, System.Drawing.FontStyle.Regular, System.Drawing.GraphicsUnit.Point, ((byte)(0)));
            this.RunButton.Location = new System.Drawing.Point(60, 147);
            this.RunButton.Name = "RunButton";
            this.RunButton.Size = new System.Drawing.Size(180, 52);
            this.RunButton.TabIndex = 2;
            this.RunButton.Text = "Run";
            this.RunButton.UseVisualStyleBackColor = true;
            this.RunButton.Click += new System.EventHandler(this.RunButton_Click);
            // 
            // OutputPictureBox
            // 
            this.OutputPictureBox.Location = new System.Drawing.Point(3, 3);
            this.OutputPictureBox.Name = "OutputPictureBox";
            this.OutputPictureBox.Size = new System.Drawing.Size(500, 500);
            this.OutputPictureBox.SizeMode = System.Windows.Forms.PictureBoxSizeMode.AutoSize;
            this.OutputPictureBox.TabIndex = 3;
            this.OutputPictureBox.TabStop = false;
            // 
            // OutputPanel
            // 
            this.OutputPanel.AutoScroll = true;
            this.OutputPanel.Controls.Add(this.OutputPictureBox);
            this.OutputPanel.Location = new System.Drawing.Point(4, 205);
            this.OutputPanel.Name = "OutputPanel";
            this.OutputPanel.Size = new System.Drawing.Size(520, 520);
            this.OutputPanel.TabIndex = 4;
            // 
            // SaveButton
            // 
            this.SaveButton.Font = new System.Drawing.Font("Microsoft Sans Serif", 20F, System.Drawing.FontStyle.Regular, System.Drawing.GraphicsUnit.Point, ((byte)(0)));
            this.SaveButton.Location = new System.Drawing.Point(280, 147);
            this.SaveButton.Name = "SaveButton";
            this.SaveButton.Size = new System.Drawing.Size(180, 52);
            this.SaveButton.TabIndex = 5;
            this.SaveButton.Text = "Save";
            this.SaveButton.UseVisualStyleBackColor = true;
            this.SaveButton.Click += new System.EventHandler(this.SaveButton_Click);
            // 
            // CardChooser
            // 
            this.AutoScaleDimensions = new System.Drawing.SizeF(6F, 13F);
            this.AutoScaleMode = System.Windows.Forms.AutoScaleMode.Font;
            this.ClientSize = new System.Drawing.Size(534, 737);
            this.Controls.Add(this.SaveButton);
            this.Controls.Add(this.RunButton);
            this.Controls.Add(this.EnterInputLabel);
            this.Controls.Add(this.InputRichTextBox);
            this.Controls.Add(this.OutputPanel);
            this.Name = "CardChooser";
            this.Text = "Card Chooser";
            ((System.ComponentModel.ISupportInitialize)(this.OutputPictureBox)).EndInit();
            this.OutputPanel.ResumeLayout(false);
            this.OutputPanel.PerformLayout();
            this.ResumeLayout(false);
            this.PerformLayout();

        }

        #endregion

        private System.Windows.Forms.RichTextBox InputRichTextBox;
        private System.Windows.Forms.Label EnterInputLabel;
        private System.Windows.Forms.Button RunButton;
        private System.Windows.Forms.PictureBox OutputPictureBox;
        private System.Windows.Forms.Panel OutputPanel;
        private System.Windows.Forms.Button SaveButton;
    }
}