Scrabble tile checking

Question

For tile checking in scrabble, you make four 5x5 grids of letters totalling 100 tiles. I would like to make one where all 40 horizontal and vertical words are valid. The set of

Answer 1

I would approach the problem (naively, to be sure) by taking a pessimistic view. I'd try to prove there was no 5x5 solution, and therefore certainly not four 5x5 solutions. To prove there was no 5x5 solution I'd try to construct one from all possibilities. If my conjecture failed and I was able to construct a 5x5 solution, well, then, I'd have a way to construct 5x5 solutions and I would try to construct all of the (independent) 5x5 solutions. If there were at least 4, then I would determine if some combination satisfied the letter count restrictions.

[Edit] Null Set has determined that there are "4,430,974 5x5 solutions". Are these valid? I mean that we have a limitation on the number of letters we can use. This limitation can be expressed as a boundary vector BV = [9, 2, 2, 4, ...] corresponding to the limits on A, B, C, etc. (You see this vector in Null Set's code). A 5x5 solution is valid if each term of its letter count vector is less than the corresponding term in BV. It would be easy to check if a 5x5 solution is valid as it was created. Perhaps the 4,430,974 number can be reduced, say to N.

Regardless, we can state the problem as: find four letter count vectors among the N whose sum is equal to BV. There are (N, 4) possible sums ("N choose 4"). With N equal to 4 million this is still on the order of 10^25---not an encouraging number. Perhaps you could search for four whose first terms sum to 9, and if so checking that their second terms sum to 2, etc.

I'd remark that after choosing 4 from N the computations are independent, so if you have a multi-core machine you can make this go faster with a parallel solution.

[Edit2] Parallelizing probably wouldn't make much difference, though. At this point I might take an optimistic view: there are certainly more 5x5 solutions than I expected, so there may be more final solutions than expected, too. Perhaps you might not have to get far into the 10^25 to hit one.

Answer 2

Here's how I would try this. First construct a prefix tree.

Pick a word and place it horizontally on top. Pick a word and place it vertically. Alternate them until exhausted options. By alternating you start to fix the first letters and eliminating lots of mismatching words. If you really do find such square, then do a check whether they can be made with those pieces.

For 5x5 squares: after doing some thinking it can't be worse than O(12000!/11990!) for random text words. But thinking about it a little bit more. Every time you fix a letter (in normal text) you eliminate about 90% (an optimistic guess) of your words. This means after three iterations you've got 12 words. So the actual speed would be

O(n * n/10 * n/10 * n/100 * n/100 * n/1000 * n/1000 ...
which for 12000 elements acts something like n^4 algorithm

which isn't that bad.

Probably someone can do a better analysis of the problem. But the search for words should still converge quite quickly.

There can be more eliminating done by abusing the infrequent letters. Essentially find all words that have infrequent letters. Try to make a matching positions for each letters. Construct a set of valid letters for each position.

For example, let's say we have four words with letter Q in it.

 AQFED, ZQABE, EDQDE, ELQUO

 this means there are two valid positionings of those:

 xZxxx
 AQFED
 xAxxx   ---> this limits our search for words that contain [ABDEFZ] as the second letter
 xBxxx
 xExxx

 same for the other

 EDQDE   ---> this limits our search for words that contain [EDLU] as the third letter
 ELQUO

 all appropriate words are in union of those two conditions

So basically, if we have multiple words that contain infrequent letter X in word S at position N, means that other words that are in that matrix must have letter that is also in S in position n.

Formula:

• Find all words that contain infrequent letter X at position 1 (next iteration 2, 3... )
• Make a set A out of the letters in those words
• Keep only those words from the dictionary that have letter from set A in position 1
• Try to fit those into the matrix (with the first method)
• Repeat with position 2

Answer 3

I'm starting with something simpler.

Here are some results so far:

   3736 2x2 solutions
8812672 3x3 solutions

The 1000th 4x4 solution is
   A A H S
   A C A I
   L A I R
   S I R E

The 1000th 5x5 solution is
   A A H E D
   A B U N A
   H U R S T
   E N S U E
   D A T E D

The 1000th 2x4x4 solution is
   A A H S | A A H S
   A B A C | A B A C
   H A I R | L E K U
   S C R Y | S T E D
   --------+--------
   D E E D | D E E M
   E I N E | I N T I
   E N O L | O V E R
   T E L T | L Y N E

Note that transposing an 'A' and a blank that is being used as an 'A' should be considered the same solution. But transposing the rows with the columns should be considered a different solution. I hope that makes sense.

Answer 4

Here are a lot of precomputed 5x5's. Left as an exercise to the reader to find 4 compatible ones :-)

http://www.gtoal.com/wordgames/wordsquare/all5

Answer 5

Final Edit: Solved! Here is a solution.

GNAWN|jOULE
RACHE|EUROS
IDIOT|STEAN
PINOT|TRAvE
TRIPY|SOLES
-----+-----
HOWFF|ZEBRA
AGILE|EQUID
CIVIL|BUXOM
EVENT|RIOJA
KEDGY|ADMAN

Here's a photo of it constructed with my scrabble set. http://twitpic.com/3wn7iu

This one was easy to find once I had the right approach, so I bet you could find many more this way. See below for methodology.

---

Construct a prefix tree from the dictionary of 5 letter words for each row and column. Recursively, a given tile placement is valid if it forms valid prefixes for its column and row, and if the tile is available, and if the next tile placement is valid. The base case is that it is valid if there is no tile left to place.

It probably makes sense to just find all valid 5x5 boards, like Glenn said, and see if any four of them can be combined. Recursing to a depth of 100 doesn't sound like fun.

Edit: Here is version 2 of my code for this.

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <stdbool.h>

typedef union node node;
union node {
    node* child[26];
    char string[6];
};

typedef struct snap snap;
struct snap {
    node* rows[5];
    node* cols[5];
    char tiles[27];
    snap* next;
};

node* root;
node* vtrie[5];
node* htrie[5];
snap* head;

char bag[27] = {9,2,2,4,12,2,3,2,9,1,1,4,2,6,8,2,1,6,4,6,4,2,2,1,2,1,2};
const char full_bag[27] = {9,2,2,4,12,2,3,2,9,1,1,4,2,6,8,2,1,6,4,6,4,2,2,1,2,1,2};
const char order[26] = {16,23,9,25,21,22,5,10,1,6,7,12,15,2,24,3,20,13,19,11,8,17,14,0,18,4};

void insert(char* string){
    node* place = root;
    int i;
    for(i=0;i<5;i++){
        if(place->child[string[i] - 'A'] == NULL){
            int j;
            place->child[string[i] - 'A'] = malloc(sizeof(node));
            for(j=0;j<26;j++){
                place->child[string[i] - 'A']->child[j] = NULL;
            }
        }
        place = place->child[string[i] - 'A'];
    }
    memcpy(place->string, string, 6);
}

void check_four(){
    snap *a, *b, *c, *d;
    char two_total[27];
    char three_total[27];
    int i;
    bool match;
    a = head;
    for(b = a->next; b != NULL; b = b->next){
        for(i=0;i<27; i++)
            two_total[i] = a->tiles[i] + b->tiles[i];
        for(c = b->next; c != NULL; c = c->next){
            for(i=0;i<27; i++)
                three_total[i] = two_total[i] + c->tiles[i];
            for(d = c->next; d != NULL; d = d->next){
                match = true;
                for(i=0; i<27; i++){
                    if(three_total[i] + d->tiles[i] != full_bag[i]){
                        match = false;
                        break;
                    }
                }
                if(match){
                    printf("\nBoard Found!\n\n");
                    for(i=0;i<5;i++){
                        printf("%s\n", a->rows[i]->string);
                    }
                    printf("\n");
                    for(i=0;i<5;i++){
                        printf("%s\n", b->rows[i]->string);
                    }
                    printf("\n");
                    for(i=0;i<5;i++){
                        printf("%s\n", c->rows[i]->string);
                    }
                    printf("\n");
                    for(i=0;i<5;i++){
                        printf("%s\n", d->rows[i]->string);
                    }
                    exit(0);
                }
            }
        }
    }
}

void snapshot(){
    snap* shot = malloc(sizeof(snap));
    int i;
    for(i=0;i<5;i++){
        printf("%s\n", htrie[i]->string);
        shot->rows[i] = htrie[i];
        shot->cols[i] = vtrie[i];
    }
    printf("\n");
    for(i=0;i<27;i++){
        shot->tiles[i] = full_bag[i] - bag[i];
    }
    bool transpose = false;
    snap* place = head;
    while(place != NULL && !transpose){
        transpose = true;
        for(i=0;i<5;i++){
            if(shot->rows[i] != place->cols[i]){
                transpose = false;
                break;
            }
        }
        place = place->next;
    }
    if(transpose){
        free(shot);
    }
    else {
        shot->next = head;
        head = shot;
        check_four();

    }
}

void pick(x, y){
    if(y==5){
        snapshot();
        return;
    }
    int i, tile,nextx, nexty, nextz;
    node* oldv = vtrie[x];
    node* oldh = htrie[y];
    if(x+1==5){
        nexty = y+1;
        nextx = 0;
    } else {
        nextx = x+1;
        nexty = y;
    }
    for(i=0;i<26;i++){
        if(vtrie[x]->child[order[i]]!=NULL &&
           htrie[y]->child[order[i]]!=NULL &&
           (tile = bag[i] ? i : bag[26] ? 26 : -1) + 1) {
                vtrie[x] = vtrie[x]->child[order[i]];
                htrie[y] = htrie[y]->child[order[i]];
                bag[tile]--;

                pick(nextx, nexty);

                vtrie[x] = oldv;
                htrie[y] = oldh;
                bag[tile]++;
           }
    }
}

int main(int argc, char** argv){
    root = malloc(sizeof(node));
    FILE* wordlist = fopen("sowpods5letters.txt", "r");
    head = NULL;
    int i;
    for(i=0;i<26;i++){
        root->child[i] = NULL;
    }
    for(i=0;i<5;i++){
        vtrie[i] = root;
        htrie[i] = root;
    }

    char* string = malloc(sizeof(char)*6);
    while(fscanf(wordlist, "%s", string) != EOF){
        insert(string);
    }
    free(string);
    fclose(wordlist);
    pick(0,0);

    return 0;
}

This tries the infrequent letters first, which I'm no longer sure is a good idea. It starts to get bogged down before it makes it out of the boards starting with x. After seeing how many 5x5 blocks there were I altered the code to just list out all the valid 5x5 blocks. I now have a 150 MB text file with all 4,430,974 5x5 solutions.

I also tried it with just recursing through the full 100 tiles, and that is still running.

Edit 2: Here is the list of all the valid 5x5 blocks I generated. http://web.cs.sunyit.edu/~levyt/solutions.rar

Edit 3: Hmm, seems there was a bug in my tile usage tracking, because I just found a block in my output file that uses 5 Zs.

COSTE
ORCIN
SCUZZ
TIZZY
ENZYM

Edit 4: Here is the final product.

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <stdbool.h>

typedef union node node;
union node {
    node* child[26];
    char string[6];
};

node* root;
node* vtrie[5];
node* htrie[5];
int score;
int max_score;

char block_1[27] = {4,2,0,2, 2,0,0,0,2,1,0,0,2,1,2,0,1,2,0,0,2,0,0,1,0,1,0};//ZEBRA EQUID BUXOM RIOJA ADMAN
char block_2[27] = {1,0,1,1, 4,2,2,1,3,0,1,2,0,1,1,0,0,0,0,1,0,2,1,0,1,0,0};//HOWFF AGILE CIVIL EVENT KEDGY
char block_3[27] = {2,0,1,1, 1,0,1,1,4,0,0,0,0,3,2,2,0,2,0,3,0,0,1,0,1,0,0};//GNAWN RACHE IDIOT PINOT TRIPY
                                                                            //JOULE EUROS STEAN TRAVE SOLES
char bag[27] =     {9,2,2,4,12,2,3,2,9,1,1,4,2,6,8,2,1,6,4,6,4,2,2,1,2,1,2};
const char full_bag[27] = {9,2,2,4,12,2,3,2,9,1,1,4,2,6,8,2,1,6,4,6,4,2,2,1,2,1,2};
const char order[26] = {16,23,9,25,21,22,5,10,1,6,7,12,15,2,24,3,20,13,19,11,8,17,14,0,18,4};
const int value[27] = {244,862,678,564,226,1309,844,765,363,4656,909,414,691,463,333,687,11998,329,218,423,536,1944,1244,4673,639,3363,0};

void insert(char* string){
    node* place = root;
    int i;
    for(i=0;i<5;i++){
        if(place->child[string[i] - 'A'] == NULL){
            int j;
            place->child[string[i] - 'A'] = malloc(sizeof(node));
            for(j=0;j<26;j++){
                place->child[string[i] - 'A']->child[j] = NULL;
            }
        }
        place = place->child[string[i] - 'A'];
    }
    memcpy(place->string, string, 6);
}

void snapshot(){
    static int count = 0;
    int i;
    for(i=0;i<5;i++){
        printf("%s\n", htrie[i]->string);
    }
    for(i=0;i<27;i++){
            printf("%c%d ", 'A'+i, bag[i]);
    }
    printf("\n");
    if(++count>=1000){
        exit(0);
    }
}


void pick(x, y){
    if(y==5){
        if(score>max_score){
            snapshot();
            max_score = score;
        }
        return;
    }
    int i, tile,nextx, nexty;
    node* oldv = vtrie[x];
    node* oldh = htrie[y];
    if(x+1==5){
        nextx = 0;
        nexty = y+1;
    } else {
        nextx = x+1;
        nexty = y;
    }
    for(i=0;i<26;i++){
        if(vtrie[x]->child[order[i]]!=NULL &&
           htrie[y]->child[order[i]]!=NULL &&
           (tile = bag[order[i]] ? order[i] : bag[26] ? 26 : -1) + 1) {
                vtrie[x] = vtrie[x]->child[order[i]];
                htrie[y] = htrie[y]->child[order[i]];
                bag[tile]--;
                score+=value[tile];

                pick(nextx, nexty);

                vtrie[x] = oldv;
                htrie[y] = oldh;
                bag[tile]++;
                score-=value[tile];
           }
    }
}

int main(int argc, char** argv){
    root = malloc(sizeof(node));
    FILE* wordlist = fopen("sowpods5letters.txt", "r");
    score = 0;
    max_score = 0;
    int i;
    for(i=0;i<26;i++){
        root->child[i] = NULL;
    }
    for(i=0;i<5;i++){
        vtrie[i] = root;
        htrie[i] = root;
    }
    for(i=0;i<27;i++){
        bag[i] = bag[i] - block_1[i];
        bag[i] = bag[i] - block_2[i];
        bag[i] = bag[i] - block_3[i];

        printf("%c%d ", 'A'+i, bag[i]);
    }

    char* string = malloc(sizeof(char)*6);
    while(fscanf(wordlist, "%s", string) != EOF){
        insert(string);
    }
    free(string);
    fclose(wordlist);
    pick(0,0);

    return 0;
}

After finding out how many blocks there were (nearly 2 billion and still counting), I switched to trying to find certain types of blocks, in particular the difficult to construct ones using uncommon letters. My hope was that if I ended up with a benign enough set of letters going in to the last block, the vast space of valid blocks would probably have one for that set of letters.

I assigned each tile a value inversely proportional to the number of 5 letter words it appears in. Then, when I found a valid block I would sum up the tile values, and if the score was the best I had yet seen, I would print out the block.

For the first block I removed the blank tiles, figuring that the last block would need that flexibility the most. After letting it run until I had not seen a better block appear for some time, I selected the best block, and removed the tiles in it from the bag, and ran the program again, getting the second block. I repeated this for the 3rd block. Then for the last block I added the blanks back in and used the first valid block it found.