KaleidoDigit

A Number Pattern Game

by Bill Price

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Click on the diagram for a javascript version of KaleidoDigit


KaleidoDigit is a number-pattern game that I invented in 1966. Although at first glance it may closely resemble “magic squares”, it is actually something quite different.

The initial position is depicted above. All numbers are in order from top to bottom. All odd numbers are in white, and all even numbers are in red.

Starting from this initial position the squares are rearranged by fixed movements. The bottom right square (#16) remains stationary for each movement.

After each movement the number pattern changes. By combining various movements these number patterns change and evolve in interesting ways, much like the mutating patterns of a kaleidoscope.

In addition to the number patterns, the color patterns within the matrix can also undergo various interesting changes, depending on the type of movement used.

There is no purpose to this “game” other than exploring the seemingly infinite variety of number and color patterns that can be produced.


The Eight Movements

There are eight movements for KaleidoDigit. Each movement has a name, but for convenience they are all designated by one-letter abbreviations. These movements are “Reverse” (R), “Odd” (O), “Even” (E), “Squares” (S), “Zigzag” (Z), “Flip” (F), “Perpendicular” (P), and “Atom” (A).

These eight movements can be classified as “weak” or “strong”. Generally speaking, the “weak” movements perform variations on a single theme, whereas the “strong” movements make radical changes to the theme itself. In most cases the most pleasing number patterns are generated when only one of the “strong” movements is executed. Combining iterations of the “strong” movements generally result in a chaotic arrangement of digits that appear to have no readily discernable pattern.

The “strong” movements are P and A. These two movements not only bring radical changes to the number pattern, but they also introduce distinct color patterns to the matrix. Without the P and A movements, the color patterns on the matrix remain unremarkable.

The “weak” movements are R, E, O, Z, S, and F. Combinations of these movements introduce a seemingly infinite variety of patterns into the basic number scheme of the matrix.

All eight movements have a “cycle life”. The cycle life of a movement is the number of iterations the movement can be executed from the initial matrix until the initial matrix is repeated.


The “Weak” Movements

Reverse (R)

This is one of the simplest of the movements. As its name suggests, it is a reversal of digits, specifically, the reversal of the first and third rows:

The cycle life of R is 2.


Odd (O)

The O movement isolates the odd and even numbers on separate rows. The first row has the first four odd digits, and the third row has the last four. The even digits are arranged in sequence on the second and last rows.

The cycle life of O is 3.


Even (E)

The E movement is identical to the O movement, but with the first and second rows exchanged. The first row has the first four even numbers, and the last row has the last four. The odd numbers are arranged in sequence on the second and third rows.

The cycle life of E is 6.


Zigzag (Z)

With the Z movement the horizontal rows are modified into a zigzag pattern. The first two rows are intertwined by switching places with the second and fourth digits, and the last two rows are similarly intertwined by switching places with the first and third digits.

The cycle life of Z is 2.


Squares (S)

The S movement takes the four horizontal rows of the matrix and curls them up into four symmetrical corners as follows:

The cycle life of S is 4.


Flip (F)

The F movement flips the entire matrix on an axis defined by the diagonal running from square 1 to square 16. The basic pattern is left undisturbed.

Note that with the ROEZS movements, the digits in the first two rows remain segregated from the digits in the last two rows. Similarly, the last two rows of digits always remain in the bottom half of the matrix. The F movement, although not disturbing the basic pattern of the matrix, introduces an intermingling of the top and bottom halves.

The cycle life of F is 2.


The “Strong” Movements

Perpendicular (P)

The P movement takes the first two digits of each row and arranges them vertically. This is accomplished by switching squares 2 and 5, and squares 10 and 13:

The cycle life of P is 2.


Atom (A)

The A movement rearranges the entire matrix in the following fashion: The first row is nestled horizontally into the middle four squares; the second row is split vertically, above and below the middle four squares; the third row is split horizontally to the left and to the right of the middle four squares; and the last row is distributed vertically into the four corner squares.

The cycle life of A is 20.


Combining Movements

Number Patterns

Interesting number patterns occur when successive combinations of these movements are applied to the initial matrix. In discerning the number pattern one simply traces the digits in sequence from 1 through 16. These are always divided into four sets of four: 1-4, 5-8, 9-12, and 13-16.

There are three stages of number pattern changes that can be observed.

  1. The first stage consists of variations using only the movements from the weak ROEZS group. Number patterns in this group always have the first two sets 1-4 and 5-8 in the top half of the matrix, and the last two sets 9-12 and 13-16 in the bottom half of the matrix.
  2. The second stage consists of combinations of the F movement with the ROEZS group. Since the F movement flips the matrix on a diagonal, the digits from the first two sets can appear in the bottom half of the matrix, and vice versa. This movement introduces a wider variety of number patterns into the matrix.
  3. The third stage consists of any combinations that include the strong movements A or P. These movements typically (but not always) produce "broken" number patterns, i.e., the number patterns are split between the sets. Examples of "broken" sets are (1-2-5-6, 3-4-7-8, 9-10-13-14, 11-12-15-16) and (1-2-7-8, 3-4-5-6, 9-10-15-16, 11-12-13-14). Multiple iterations of the strong movements usually introduce chaotic and indiscernible patterns into the matrix.

Typically, the most pleasing number patterns are produced by the second group, i.e., number patterns that result from the weak movements ROEZSF.

We can put movement abbreviations together into a movement string. The matrix that results from this sequence of movements can thus be named by this movement string. For instance, the matrix that results from the movements E, S, A, and O performed in sequence on the initial matrix can be designated by the string “ESAO”.


Examples of Number Patterns

Here is a very simple first-stage pattern created by the string ROE:

Since this is a first-stage number pattern, note that the first two sets 1-4 and 5-8 are in the upper half of the matrix, and the last two sets 9-12 and 13-16 are in the bottom half.

Observe that the four sets all trace a rectangular pattern. This pattern, or shape, is called the "Short Rectangle". All of the rectangles are oriented horizontally within the matrix; this is the orientation of the set. The rectangle for the first set is located in the upper left corner of the matrix. This is called the position of the set.

Note also that the digits in each set trace a "Z" pattern. The Z pattern is called the path of the set. It can also be noted that the starting points for each set are at opposite corners; i.e., the first set 1-4 starts in the upper right corner, whereas the second set starts in the lower left corner. Similarly the third set starts in the lower right corner, and the fourth set starts in the upper left corner. The starting point for each set is called the origin of the set. Movement from the origin to the second point within the first set is in a leftward or counterclockwise direction. This is called the direction of the set.

Examining this sample matrix we have identified the following six attributes for number patterns:


Classification of Shapes

Here are the 23 number patterns ("Shapes") that occur with the weak movements ROEZSF. Please note that these represent only the shapes themselves, independent of Position or Orientation within the matrix.

Row Diagonal Minisquare Mesosquare
       
Maxisquare Short Rectangle Long Rectangle Full Rectangle
       
Miniparallelogram Mesoparallelogram Maxiparallelogram Miniparallelobar
       
Mesoparallelobar Maxiparallelobar Long Diamond Full Diamond
       
Thin Diamond Minitrapezoid Mesotrapezoid Maxitrapezoid
       
 
Trapezium T Y  

Examples of ROEZSF Movement Strings

Now let us follow through some movements beginning from the initial matrix and examine the number patterns resulting from each movement.

Initial Matrix

SET SHAPE ORIENTATION POSITION ORIGIN PATH DIRECTION
1-4 Row Horizontal 1 1 U Right
5-8 Row Horizontal 2 1 U Right
9-12 Row Horizontal 3 1 U Right
13-16 Row Horizontal 4 1 U Right

 

S

SET SHAPE ORIENTATION POSITION ORIGIN PATH DIRECTION
1-4 Minisquare n/a NW SW U Left
5-8 Minisquare n/a NE SE U Right
9-12 Minisquare n/a SW NW U Right
13-16 Minisquare n/a SE NE U Left

 

SO

SET SHAPE ORIENTATION POSITION ORIGIN PATH DIRECTION
1-4 Short Rectangle Horizontal NW NE U Right
5-8 Short Rectangle Horizontal NE SE U Left
9-12 Short Rectangle Horizontal SW NW U Left
13-16 Short Rectangle Horizontal SE SW U Right

 

SOR

SET SHAPE ORIENTATION POSITION ORIGIN PATH DIRECTION
1-4 Miniparallelogram Horizontal Right N NW X Right
5-8 Miniparallelogram Horizontal Left N SE X Right
9-12 Miniparallelogram Horizontal Right S NE X Left
13-16 Miniparallelogram Horizontal Left S SW X Left

 

SORF

SET SHAPE ORIENTATION POSITION ORIGIN PATH DIRECTION
1-4 Miniparallelogram Vertical Left W NW X Left
5-8 Miniparallelogram Vertical Right W SE X Left
9-12 Miniparallelogram Vertical Left E SW X Right
13-16 Miniparallelogram Vertical Right E NE X Right

 

SORFE

SET SHAPE ORIENTATION POSITION ORIGIN PATH DIRECTION
1-4 Mesotrapezoid E W NE X Right
5-8 Mesotrapezoid W W SE X Left
9-12 Mesotrapezoid E E SE X Left
13-16 Mesotrapezoid W E NE X Right

 

SORFES

SET SHAPE ORIENTATION POSITION ORIGIN PATH DIRECTION
1-4 Full Diamond Right n/a N Z Right
5-8 Full Diamond Left n/a S Z Left
9-12 Thin Diamond Right n/a S Z Right
13-16 Thin Diamond Left n/a N Z Left

 

SORFESO

SET SHAPE ORIENTATION POSITION ORIGIN PATH DIRECTION
1-4 Long Rectangle Vertical Center NW Z Left
5-8 Long Rectangle Horizontal Center SE Z Left
9-12 Minisquare n/a Center SE Z Left
13-16 Maxisquare n/a n/a NW Z Left

 

Now let us examine a more interesting pattern created by the string FSROEZESZOEESER. Note how the digits in sequence from 1 through 16 trace a virtually unbroken leftward spiral:

SET SHAPE ORIENTATION POSITION ORIGIN PATH DIRECTION
1-4 Thin Diamond Right n/a N U Left
5-8 Full Diamond Right n/a N U Left
9-12 Full Diamond Left n/a E U Left
13-16 Thin Diamond Left n/a E U Left

 

There is a wide variety of number patterns that can be created with “KaleidoDigit”, and the above examples only give us a small sampling of the possibilities.


Color Patterns

The Weak Color Patterns

Color patterns created by the arrangement of the red and white squares (i.e., the even and odd numbers) remain rather regular and symmetrical with the weak movements. There are nine color patterns for the weak movements.

Bars Center Quarters Checkers Foursquare
       
Halves Long Checkers Quarters Sandwich
       
     
Shifted Bars      
       

Note that these nine patterns are defined independent of their vertical or horizontal orientation. The "Flip" movement changes the orientation for "Bars", "Center Quarters", "Halves", "Long Checkers", "Sandwich", and "Shifted Bars", but the designations for the color patterns stay the same.


The Strong Color Patterns

With the strong movements A and P, a wide variety of interesting color patterns arise, many of them non-symmetrical. The most common of these are named as follows:

Anvil Anvil over Staple Anvil under Staple Arrow
       
Bar in Water Barracks Bars and Checkers Big PI
       
Bowtie Box in Center Box on Side Broken Redwing
       
Broken Rods Broken Swastika Broken T C in Water
       
CI CI in Water Capital C Capital F
       
Capital I Capital P Chapel Checkered Anvil
       
Checkers and Quarters Checkers and Red Bars Checkers and Stripes Checkers and White Bars
       
Checkers on Halves Club Corner Boxes Crazy T's
       
Diamond Diamond S Double Anvil Double L
       
Double S Double T Double Z Embracing F's
       
F with Colon Floating Anvil Flying Goose Four-Leaf Clover
       
Jackrabbit Jellyfish Jewel Jewel Cross
       
Korean P's Leaky Faucet Little PI Long Bars and Checkers
       
Long Quarters Moosehead Narrow Staple and Dots Narrow Staples
       
Packed L's Piston Propeller Question Mark
       
Redwing Sandwich and Shifted Bars Shadow Anvil Shadow Cross
       
Shadow Diamond S Shadow Jewel Shadow Moosehead Shadow Piston
       
Shadow Question Marks Shadow Truck Siamese C's Siamese F's
       
Side Staples Skier Snake and Dot Spiral
       
Spiral F's Spiral Question Marks Split Anvil Stairs
       
Swastika Telephone Triple Boxes Truck
       
 
Truck in Water Wide Staple and Dots Wide Staples  
       

Note that nearly all of the above color patterns are so designated regardless of orientation within the matrix. Also, reversing the colors also has no effect on the patterns' names (the only exceptions being "Checkers and Red Bars" and its reversed counterpart "Checkers and White Bars").


Combining Number and Color Patterns

The most aesthetically pleasing patterns in KaleidoDigit occur when an interesting number pattern occurs concurrent with an equally interesting color pattern. The most desirable of these patterns are those that can be created by a minimal number of movements from the initial matrix. The most notable of these is the string FPZOR:

The color pattern is the Arrow. The attributes for the number patterns are as follows:

SET SHAPE ORIENTATION POSITION ORIGIN PATH DIRECTION
1-4 Diagonal Right n/a 4 U Left
5-8 Long Diamond Left n/a N Z Left
9-12 Mesosquare n/a NW NE U Left
13-16 Mesosquare n/a SE NE U Left

 


How Many Combinations?

How many KaleidoDigit patterns exist? Within the 16-square matrix there are 20,922,789,888,000 (16 factorial) possible arrangements of digits. For “KaleidoDigit”, however, this number is reduced to 1,307,674,368,000 (15 factorial) by the fact that the number 16 always remains in the bottom right corner. Additionally, many of these combinations are not possible to create with the eight movements available.

More interesting for “KaleidoDigit” would be the question: “How many movement strings exist that create unique matrices?” This is a challenging question, made more challenging by the fact that a single matrix can in many cases be produced by more than one string. For instance, RSSE and EOERO are identical, i.e., they create the same matrix.

I once investigated this question with a simple computer program that was capable of generating KaleidoDigit matrices and discarding duplicates. I put the following arbitrary limitations on the program:

  1. Movement strings can be from 1 to 10 characters long, but cannot exceed 10 characters.
  2. Store no more than 2000 unique matrices. If more than 2000 are found, terminate the program.
  3. Use input parameter to limit the permissible movement types.

By feeding the program only one movement type, the result is of course the cycle life of the movement itself. For example, by giving the program only the R movement, only two unique matrices can be found, namely, the initial matrix and the R matrix.

Since the program was capable of working out all possible 1 to 10-char movement strings given on the input parameter, I decided to keep all computations on a relatively small scale by examining only strings of 2 and 3 characters. This was due to the large number of computations involved. With 2 characters the program only has to work out the matrices for 2046 strings. (This is 21 + 22 + 23 ... + 210). With 3 characters there are 88,572 strings (31 + 32 + 33 ... + 310). But with 4 characters there are 1,398,100, and with all 8 characters there are 1,227,133,512!

The program was written to discard automatically any “superfluous“ movements that included the full cycle life of any one movement. For instance, for “RO” we ignore any movements that include the substring “RR” or the substring “OOO”, since these would of course be duplicates of corresponding strings that omit the “RR” or the “OOO”.

I arbitrarily decided to examine only strings containing the “weak” movement types, i.e., only the movements ROEZSF.

First we examine only two of the movements in combination. With six different movement types, we can pair them up 15 different ways: RO, RE, RS, RZ, RF, OE, OS, OZ, OF, ES, EZ, EF, SZ, SF, and ZF.

We then examine three of the movements in combination. With the six we can group them in 20 different ways: ROE, ROS, ROZ, ROF, RES, REZ, REF, RSZ, RSF, RZF, OES, OEZ, OEF, OSZ, OSF, OZF, ESZ, ESF, EZF, SZF.

The table below shows the results. The first column shows the combination of movement types allowed, the second column shows the number of unique matrices found (using up to 10 iterations of any combination of these allowable movements), the third column shows the number of matrices examined, and the last column shows the number of the last unique matrix found. I have sorted the table by number of unique matrices found.

In other words, for OEZ (i.e., using only the movements O, E, and Z, in any combination, from 1 character up to 10 characters) there were 760 unique matrices found. The program worked through 88,572 strings using the movements O, E, and Z before terminating. The last unique matrix was discovered on the 73,898th generated string.

For those combinations that produced more than 2000 unique matrices, the word “overflow” appears in column one. For example, for the the movement combination R, S, and F we found over 2000 unique matrices, the overflow occurring after the 14,598th matrix was examined.

It should also be noted that the initial matrix itself is included in each tally.

Movements Unique Matrices Matrices Examined Last Unique Matrix Found
RZ 6 2046 8
OF 12 2046 18
RF 12 2046 83
ZF 12 2046 83
OR 24 2046 163
OE 24 2046 37
RS 24 2046 89
OS 24 2046 53
ES 24 2046 53
SZ 24 2046 71
OZ 95 2046 1703
OES 96 88572 299
RSZ 96 88572 1378
RE 172 2046 2005
ROE 192 88572 1409
SF 209 2046 1704
EF 277 2046 1703
EZ 301 2046 1704
OEZ 760 88572 73898
ROF 1274 88572 80376
OZF 1319 88572 80450
ROZ 1320 88572 80969
ROS 1342 88572 65101
REZ 1344 88572 54645
OSZ 1344 88572 60521
RES 1344 88572 61706
ESZ 1344 88572 63120
RZF 1465 88572 81109
RSF overflow 14599 14598
SZF overflow 13785 13784
OSF overflow 12637 12636
OEF overflow 11689 11688
REF overflow 8968 8965
EZF overflow 8555 8553
ESF overflow 8062 8061

The least productive of the movement combinations is RZ which is capable of yielding only six different matrices. Since one of these matrices is the initial matrix, there are only five different ways of combining R and Z into a string to create a unique matrix. These five are: R, Z, ZR, RZ, and RZR. All other strings, such as ZRZ, RRZZRZR, ZRZRZR, etc., only result in matrices produced by the first five, or in the initial matrix.

Following RZ are OF, RF, and ZF, each of which can only produce twelve matrices - or rather, eleven matrices excluding the initial matrix.

For OF these eleven are:
O, F, OO, FO, OF, FOO, OFO, OOF, FOF, OFOO, and OOFO.

For RF they are:
R, F, FR, RF, RFR, FRF, FRFR, RFRF, RFRFR, FRFRF, and FRFRFR.

For ZF they are:
Z, F, FZ, ZF, ZFZ, FZF, FZFZ, ZFZF, ZFZFZ, FZFZF, and FZFZFZ.

Next come OR, OE, RS, OS, ES, and SZ, each with 24 (i.e., 23). These are as follows:

OR:
R, O, OR, RO, OO, ROR, OOR, ORO, ROO, OROR, ROOR, RORO, OORO, OROO, ROROR, OOROR, OROOR, ROORO, ROROO, ROOROR, ROROOR, OROORO, and OROOROR.

OE:
O, E, OO, EO, OE, EE, EOO, OEO, EEO, OOE, EOE, OEE, EEE, OEOO, EEOO, OOEO, EOEO, OEEO, EEEO, OOEOO, EOEOO, OEEOO, and EEEOO.

RS:
R, S, SR, RS, SS, RSR, SSR, SRS, RSS, SSS, SRSR, RSSR, SSSR, SSRS, SRSS, SSRSR, SRSSR, RSSRS, SSSRS, SSRSS, RSSRSR, SSSRSR, and SSRSSR.

OS:
O, S, OO, SO, OS, SS, SOO, OSO, SSO, OOS, SOS, SSS, OSOO, SSOO, OOSO, SSSO, SOOS, SSOS, SSSOO, SOOSO, OSOOS, SSOOS, and SSSOS.

ES:
E, S, EE, SE, ES, SS, SEE, ESE, SSE, EES, SES, SSS, ESEE, SSEE, EESE, SESE, SSSE, ESES, SSES, SESEE, SSESE, EESES, and SSSES.

SZ:
S, Z, SS, ZS, SZ, SSS, ZSS, SZS, SSZ, ZSZ, ZSSS, SZSS, SSZS, ZSZS, ZSSZ, SZSSS, SSZSS, ZSZSS, ZSSZS, SZSSZ, SSZSSS, ZSZSSS, and ZSSZSS.

The most fruitful combinations of movements are RSF, SZF, OSF, OEF, REF, EZF, and ESF, each of which produces over 2000 unique matrices with strings up to 10 characters. Of these ESF reached an overflow condition before any of its “competitors”, suggesting that ESF is the most productive combination of movements within the ROEZSF group.


How Many Variations of One Set?

It is considerably easier to calculate the number of possible variations of one set with the weak movements ROEZSF within the matrix. Here we simply add together the products of the number of possible attributes for each shape and eliminate all the impossible combinations.

Recall that each shape has five attributes: orientation, position, origin, path, and direction.

The following is a tally of the possible combinations of shape, orientation, and position:

Row

8
Diagonal
2
Minisquare
9
Mesosquare
4
Maxisquare
1
Short Rectangle
12
Long Rectangle
6
Full Rectangle
4
Miniparallelogram
12
Mesoparallelogram
8
Maxiparallelogram
4
Miniparallelobar
12
Mesoparallelobar
8
Maxiparallelobar
4
Long Diamond
2
Full Diamond
2
Thin Diamond
2
Minitrapezoid
12
Mesotrapezoid
8
Maxitrapezoid
4
Trapezium
8
T
4
Y
4
Total:
140

This means that there are 140 different ways of shaping, positioning, and orienting one set within the matrix. (This of course ignores the fact that "16" always has a fixed point in the matrix, an issue that we address below.)

Moving on to the last three attributes:

Therefore we have:

Row

8 x 24 =192
Diagonal
2 x 24 = 48
Minisquare
9 x 24 = 216
Mesosquare
4 x 24 = 96
Maxisquare
1 x 24 = 24
Short Rectangle
12 x 24 = 288
Long Rectangle
6 x 24 = 144
Full Rectangle
4 x 24 = 96
Miniparallelogram
12 x 24 = 288
Mesoparallelogram
8 x 24 = 192
Maxiparallelogram
4 x 24 = 96
Miniparallelobar
12 x 24 = 288
Mesoparallelobar
8 x 24 = 192
Maxiparallelobar
4 x 24 = 96
Long Diamond
2 x 24 = 48
Full Diamond
2 x 24 = 48
Thin Diamond
2 x 24 = 48
Minitrapezoid
12 x 24 = 288
Mesotrapezoid
8 x 24 = 192
Maxitrapezoid
4 x 24 = 96
Trapezium
8 x 24 = 192
T
4 x 24 = 96
Y
4 x 24 = 96
Grand Total:
3360

What this says, for instance, is that there are 3,360 different and unique ways of arranging the set 1,2,3,4 in the matrix.

However, we are not done yet, since we need to eliminate all of the combinations that have the number "16" in any but the lower right-hand square of the matrix. We can do this as follows:

First we make a tally of all the shapes that can be positioned and oriented to include the "16" square:

Row

2
Diagonal
1
Minisquare
1
Mesosquare
1
Maxisquare
1
Short Rectangle
2
Long Rectangle
2
Full Rectangle
2
Miniparallelogram
2
Mesoparallelogram
2
Maxiparallelogram
2
Miniparallelobar
2
Mesoparallelobar
2
Maxiparallelobar
2
Long Diamond
0
Full Diamond
0
Thin Diamond
2
Minitrapezoid
2
Mesotrapezoid
2
Maxitrapezoid
2
Trapezium
2
T
1
Y
1
Total:
35

So there are 35 unique ways of positioning and orienting a shape using the "16" square.

We know that there are 24 different permutations of each of these for origin, path, and direction. Out of these 24 there are only 6 that have "16" sequenced last. That means that there are 18 in each set of 24 permutations that must be eliminated. To illustrate this let us examine the 24 permutations of a Minisquare positioned SE, i.e., in the lower right-hand corner:

11 12 16 15 <== eliminate
11 12 15 16
11 15 12 16
11 16 12 15 <== eliminate
11 15 16 12 <== eliminate
11 16 15 12 <== eliminate
12 11 16 15 <== eliminate
12 11 15 16
12 15 11 16
12 16 11 15 <== eliminate
12 15 16 11 <== eliminate
12 16 15 11 <== eliminate
15 11 16 12 <== eliminate
15 11 12 16
15 12 11 16
15 16 11 12 <== eliminate
15 12 16 11 <== eliminate
15 16 12 11 <== eliminate
16 11 15 12 <== eliminate
16 11 12 15 <== eliminate
16 12 11 15 <== eliminate
16 15 11 12 <== eliminate
16 12 15 11 <== eliminate
16 12 12 11 <== eliminate

Multiplying our above total of 35 by 18 we obtain 630.

That means that from our above grand total of 3360 we must subtract 630. The result is 2730.

Therefore the total number of possible combinations for a set is 2,730. However, this number can be a bit misleading since it includes combinations with the set 13-14-15-16 (which always has the "16" in the lower right-hand corner), along with all other sets (which never touch the lower right-hand corner). We should really separate the number 2,730 into these two distinct and mutually exclusive groups.

Obtaining the number of variations within the 13-14-15-16 set is easily done by multiplying the above 35 by 6, since 6 out of every 24 combinations of 13-14-15-16 have 16 in the last position. This total is 210. Subtracting 210 from 2730 we obtain 2520, the number of combinations possible for the other sets.

In conclusion:


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Copyright 1966, 2006 by Bill Price