This report introduces variants of the popular game of bingo as played in the United States. Section 1 is a brief review of traditional bingo, and new material begins in section 2.
Section 1. An ordinary bingo card is pictured in figure 1. Players often use several at once, but all cards are reckoned independently.
figure 1 

There are 5 columns, where:
The master of ceremonies (more briefly, the caller) has a container with 75 tokens each bearing a number between 1 and 75 preceded by the corresponding letter. For instance, there is a token that reads "G59". The caller draws tokens randomly, one by one, and announces to all players the letter and number on each token as it is drawn. Alternatively, equivalent electronic equipment can be used.
Each player marks a number on their card when it is announced. Meanwhile, the free square is assumed to have already been marked on all cards. The letters in the top row are never marked; they are purely informative. After enough numbers have been called, some player(s) will have completed a prescribed pattern, announcing "bingo!" and claiming a prize. This brings the game to an end. Ties may occur, athough a procedure that will usually break them is explained in section 3.
Variations abound, but here are the 12 standard patterns:
When a player has completed a winning pattern, some irrelevant squares will probably have been marked, but those are ignored.
Two characteristics of traditional bingo are pertinent here:
It is best if there are never two identical (or nearly identical) cards in play. Fortunately, it is mathematically possible to generate millions of cards that are quite dissimilar. On the other hand, a proprietor might allow players to design their own cards, placing their favorite numbers in freelychosen locations; such players gain no advantage, but might enjoy the game more. Although ties can occur, if the winning patterns are defined in advance it may be possible to generate a large set of bingo cards that will never lead to a tie in no matter what sequence the numbers are called.
Section 2. Figure 2A1 is an example of what a 37cell card, in a hexagonal grid, could look like. The center cell is free, and the other cells hold numbers 1 through 36 in random locations.
figure 2A1 — 37 cells 

Unlike traditional bingo, every card contains every number, meaning that every player can mark a cell at every call. As a result, players might feel more "productive". In a game using this kind of card, the caller would consequently have 36 tokens to draw from.
Figures 2A2 and 2A3 are smaller and larger alternatives.
figure 2A2 — 19 cells  figure 2A3 — 61 cells 

With as many as 37 cells, it might be timeconsuming for a player to find the right one after each call, hindering the pace of the game. For that reason, numbers can be organized into zones, printed in different colors for conspicuity. When this is done, the numbers used for a 37cell card will not be 1…36, but rather be 36 twodigit values from the following six ranges:
10…15
20…25
30…35
40…45
50…55
60…65
Figure 2B has four examples of 37cell zoned cards; all could be in use simultaneously in a bingo game. There is no particular relation between colors and numbers, except that adjoing zones are of different colors. There are are 6! × 6! = 720 × 720 = 518,400 different cards available in this format.
figure 2B — 37 cells  

figure 2B1  figure 2B2 
figure 2B3  figure 2B4 
6 zones of 6 cells, 1 free cell 
From one card to the next, each zoneful of numbers might be in a different location. For instance, the range 40…45 appears in the southwest zone of 2B1, the northeast zone of 2B2, the southeast of 2B3, and the northwest of 2B4. In a large set of uniformly randomized cards, each number will appear in each of the 36 possible locations an approximately equal number of times. This differs from conventional bingo, where the range 1…15 is always in the first column, 16…30 in the second, et cetera.
Zoning does not affect the rules, strategy, or probabilities of the game in any way; it merely aids in finding numbers faster.
With zoning, the 61cell card becomes more practicable. Here are two examples:
figure 2C — 61 cells  

6 zones of 10 cells, 1 free cell 
Zoning brings an even larger card into the realm of consideration:
figure 2D — 91 cells 

9 zones of 10 cells, 1 free cell 
The cards illustrated so far have had the overall shape of a regular hexagon, but alternatives will appear in section 4.
Some players may prefer an alternate orientation of the grid, as in figure 2E:
Figure 2E  Figure 2B1 repeated for comparison 

equivalent: 6 zones of 6 cells, 1 free cell 
Finally, it is sometimes helpful in explanations to use a reference grid, numbered straight through:
figure 2F 

Comment. The free cell is often indispensible for constructing zones. For instance, the cards of figures 2A1, 2A2, and 2A3 have 37, 19, and 61 cells repectively. All of these are prime numbers, which fact precludes a partition into zones of equal size. Designating one cell as free reduces the numbers to 36, 18, and 60; and these have nontrivial factorizations, making zones of equal size possible. Even with some of the nonhexagonal cards starting in section 4 below, this consideration comes into play.
Section 3A. The patterns that qualify as winners can be anything that the participants can agree on. Figure 3A contains some possible floating patterns, in other words patterns that can appear anywhere on the card, at least if the card is large enough.
figure 3A  

5straight, 6straight, 7straight two other rotations  7wye one other rotation 
5sharp, 7sharp five other rotations  5blunt, 7blunt five other rotations 
6triangle, 9triangle, 10triangle one other rotation  6hexagon, 7hexagon no other rotations 
Some players might opt to recognize the 9triangle, but only as long as the center remains empty. In other words, a 10triangle would not qualify as a 9triangle. Similar remarks apply to the 6 and 7hexagons. Note that if a pattern can be ruined when the numbers of extraneous cells are called, there might sometimes be a game that nobody wins.
Section 3B. Here is an example of play and tiebreaking in a game where the winning pattern is a 5straight. So far, the following 12 numbers have been called, but not necessarily in this order: 15, 16, 34, 37, 38, 41, 47, 54, 58, 59, 67, and 68. In figure 3B those cells are drawn in a lighter color, as is the free cell. Neither player has yet completed a 5straight.
figure 3B  

player 1  player 2 
The next number called is 51, which simultaneously completes the pattern (drawn in white in figure 3C) for both players.
figure 3C  

player 1 — 4734385159  player 2 — 5158541568 
Who wins? The tie is broken in favor of whichever player's pattern contains the highest number. For player 1, that is 59; player 2, 68; so player 2 wins. If highest numbers are equal, then secondhighest are compared, and so forth. A free cell counts zero, and cells that are not part of the potentially pattern are ignored. By this rule, which can also be applied to traditional bingo, nearly every tie can be resolved.
Section 3C. Multiple patterns can be recognized within the same game. For instance, suppose the winning patterns in a game are:
After a player completes a pattern and claims a prize, play continues until all other prizes have been awarded. A key interpretation is that a longer straight contains all shorter straights. Here are the cases:
If the first claim is for a 7straight …  … that player wins $7:
 
If the first claim is for a 6straight …  … that player wins $3:
 
If the first claim is for a 5straight …  … that player wins $1.  If the next claim is for a 7straight …  … that player wins $6:

If the next claim is for a 6straight …  … that player wins $2.
Play continues until someone claims a 7straight and wins $4. The game ends.  
The total of prizes will always be $7. 
If multiple players make claims at the same time, anyone with a 7straight has priority over anyone with a 6straight; and 6straight over 5straight. Cell numbers are not considered unless the lengths of straights are equal. Example:
In a game where no prize has yet been awarded, Anne and Bill simultaneously shout "bingo". Anne has a 6straight with 272524201915. Bill has a 5straight with 5958514642. Anne has priority over Bill because she has the longest straight, even though Bill's 5straight has higher numbers than either 5straight residing within Anne's 6straight. Anne wins $3 and Bill wins $0.
Section 3D. Another pattern that might be used is the skip. On the 37cell card, a skip of length 4 will run from edge to edge. There are 12 possible instances, 4 in each of 3 directions. In figure 3D1 are highlighted the examples 1112434, 10182633, and 15202529. Cells that a 4skip can never reach are marked with an 'X'.
figure 3D1 — 37 cells 

Figure 3D2 highlights some 3skips, and 3D3 some 5skips.
figure 3D2 — 19 cells  figure 3D3 — 61 cells 

Some players may choose to allow a skip of less than maximal length, such as a 4skip on a 61cell card. In this case, the 'X' cells become useful. On the other hand, if skips must always be maximal, the 'X' cells might be deleted entirely, reducing the 37cell card to 30 cells, and the 61cell to 55.
Section 4. By way of example, here are some cards that are not regular hexagons. As for zoning, twodigit numbers remain workable if there are at most 9 zones, each containing at most 10 cells, as figure 2D. All zones on a card should have the same number of cells.
Figure 4A is derived by removing the top, lower left, and lower right rows of figure 2D. This requires adjustment of the numbers.
figure 4A — 73 cells 

9 zones of 8 cells, 1 free cell 
A second truncation produces figure 4B.
figure 4B — 52 cells  

3 zones of 17 cells, 1 free cell  7 zones of 7 cells, 3 free cells 
Expanding figure 4B into a triangle yields 4C in two varieties. Because these two cards have different numbers to some extent, they could not be used in the same game.
figure 4C — 55 cells  

9 zones of 6 cells, 1 free cell  6 zones of 9 cells, 1 free cell 
Figure 4D, neither a hexagon nor a triangle, contains two free cells. A possible tiebreaking rule is that a 6straight that uses neither free cell ("the hard way") has priority over a 6straight that uses one of them, which in turn has priority over the 6straight that uses both.
figure 4D — 50 cells  

6 zones of 8 cells  8 zones of 6 cells, 2 free cells 
Instead of a free cell, figure 4E has a hole in the middle. A proposed winning pattern is any loop that surrounds the hole, as pictured on the right. A possible tiebreaking criterion is to give preference to the player with the shortest loop; pictured is 20 cells, while the minimum is 9.
figure 4E — 63 cells  

7 zones of 9 cells, 0 free cells 
In figure 4F, the task is to connect the top row (638188 etc) to the bottom row (656283 etc). Two examples are shown.
figure 4F — 72 cells 

6 zones of 8 cells, 0 free cells 
Figure 4G is a hexagon with a large hole.
figure 4G — 72 cells 

8 zones of 9 cells, 0 free cells 
Section 5. Figure 5A displays a card with numerous scattered holes. These noncells are lettered A through M for reference, with F skipped because that letter would represent a free cell.
figure 5A — 48 cells 

6 zones of 8 cells, 0 free cells 
The basic winning pattern is the ring, meaning the 6 cells that immediately surround any hole. Adjacent to hole H, for example, are the cells 414757545016.
Some players might go further and specify that the winning pattern contain two rings, rather than one. An optional constraint here is that the two rings have a common cell. For instance, rings D and E (11 cells total) share cell 34, and would qualify. On the other hand, rings J and L (12 cells total) have no cell in common and would not meet that requirement.
Beyond that, there can be a pattern of three rings any two of which must share a cell; this may work better on a large card. Rings A, C and D (15 cells total) form an example: A and C share cell 40; A and D, cell 46; C and D, cell 42. The holes form an equilateral triangle.
There is a different way to play that allows a player to use judgement. Under a possible scenario, a single ring (6 cells) wins a prize of $1, a double ring (11 cells) pays $2, and a triple (15 cells) wins $4. However, when a player claims a prize, that card is retired for the rest of that game and cannot win any further prize. Of course, that player may continue to compete with their other cards. After all prizes have been awarded, and a new game begins, any retired cards come back into play.
To illustrate, a player who completes a single ring has a choice:
Much as in section 3C, if the first claimant has a double ring, they win $3. Anyone who claims a triple wins all outstanding prizes.
This kind of card also works with straights instead of rings; a suggestion is to recognize any straight that runs from edge to edge (is maximal). A straight that itself is on the edge of the card is termed exterior, otherwise interior.
examples of maximal straights
see figure 5A  
exterior  interior  

4straight  44636264 2 others  none 
6straight  515653232224 2 others  323031262723 2 others 
8straight  none  1312505455262025 5 others 
The first player to claim a straight in any of the four categories wins a prize, and the game continues until all prizes have been claimed.
This multipleprize situation differs from that of section 3C in an important way. There, a larger straight contains a smaller straight. Here, however, the four categories of straight, being distinguished as exterior versus interior, are disjoint. Thus it is possible for two players to win prizes on the same call, or one player to win two separate prizes.
Figures 5B and 5C show cards intended for the 9triangle pattern. Enough cells are provided so that any hole can be surrounded by a 9triangle pointed either up or down. Also, every cell can be a part of at least one 9triangle. (Edgetoedge straights might be recognized instead, much as above.)
figure 5B — 42 cells 

6 zones of 7 cells, 0 free cells 
In previous drawings, three colors (red, green, blue) were enough to render adjacent zones in different colors. In the larger card of figure 5C, four colors (adding yellow) become necessary, although this does not affect the play of the game. As long each zone is contiguous, four colors will be enough.
figure 5C — 63 cells 

9 zones of 7 cells, 0 free cells 
Figure 5D has the same shape as 5C, but includes 3 free cells.
figure 5D — 63 cells 

6 zones of 10 cells, 3 free cells 
Section 6. Figure 6A introduces the triangular grid as an alternative to the hexagonal. Except at the edges, each triangular cell has 3 sidewise neighbors and 9 cornerwise neighbors, while each hexagonal cell has 6 sidewise and 0 cornerwise. This means that the two grids are substantively different, even though their angles are all multiples of 60 degrees. Figure 2B1, a rough counterpart to 6A, is redrawn here in order to make the differences more conspicuous.
figure 6A — 37 cells  figure 2B1 — 37 cells
repeated for comparison 

6 zones of 6 cells, 1 free cell 
Figure 6B shows some winning patterns that might be chosen. Note that a straight of odd length comes in one variety; but a straight of even length comes in two, which are reflections of each other. This sort of behavior is not seen with the hexagonal grid.
figure 6B  

5straight, 7straight five other rotations  6hexagon no other rotations  12ring one other rotation  
 
two different 6straights two other rotations  9triangle one other rotation  12star no other rotations  6bead, 8bead two other rotations 
Figure 6C is a large triangle with a hole in the middle, resembling figure 4E.
figure 6C — 60 cells 

6 zones of 10 cells, 0 free cells 
Figure 6D has many small holes, something like figure 5A. It is well suited to one of the two rotations of the 12ring.
figure 6D — 63 cells 

9 zones of 7 cells, 0 free cells 
Section 7. Figure 7A shows another kind of grid, this one with hexagons, squares, and triangular holes. Although the layout is superficially similar to the plain hexagonal grid of figure 2C1, they are not equivalent. For instance, the 5straight 3139646368 on 2C1 lacks a counterpart on 7A, because holes A, B, C, and D interrupt the pattern.
figure 7A — 61 cells
19 hexagons, 42 squares  figure 2C1 — 61 cells
repeated for comparison 
6 zones of 10 cells, 1 free cell 
In figure 7B are some winning patterns that might be chosen. Each has at least one hexagon, and at least one square.
figure 7B  

two kinds of 5straight two other rotations  7star no other rotations 
4straight, 6straight five other rotations  6ring one other rotation 
12ring no other rotations  double 6ring, 9 cells two other rotations  double 6ring, 11 cells two other rotations 
If all the winning patterns recognized in a game use hexagons exclusively, the squares are relegated to being a useless distraction. Similarly, the hexagons are "wasted" if squares are recognized exclusively. Hence those choices are not recommended. Of course, if some patterns are pure hexagons while other patterns are pure squares, this problem is avoided.
Section 8. The cards in figure 8A are based on the same tessellation as those in section 7, but now the hexagons have become holes, and the triangular holes have become cells. Two candidate zoning patterns are shown.
figure 8A  

9 zones of 6 cells, 0 free cells  6 zones of 9 cells, 0 free cells 
An obvious choice for the winning pattern would be a ring of twelve cells, as around hole C in the left diagram: 609590939491252223246365.
Besides the usual pergame prizes, this card design is suitable for special prizes that apply to a series of games: The first player to complete a ring around hole A gets an extra prize, the first around hole B, et cetera, for a total of 7 extra prizes.
Section 9. Another grid uses octagons and squares, and does not involve any holes. The two cards in figure 9A are similar, but not equivalent unless winning patterns are selected in a rather contrived way.
figure 9A  

8 zones of 6 cells, 1 free cell 
Here are some possible winning patterns:
figure 9B  

 
two kinds of 5straight two other rotations  major 8ring no other rotations  minor 5cross no other rotations 
 
4straight, 6straight five other rotations  minor 8ring no other rotations  major 5cross no other rotations 
Colophon. The pages were prepared on a Macintosh computer running OS X version 10.11. The HTML was written with TextEdit. The images are rendered using Scalable Vector Graphics generated by a C++ program running within Xcode.