Hexagonal bingo.
Version of Wednesday 9 January 2019.
Dave Barber's other pages.

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:

• column "B" contains 5 numbers randomly selected, without repeats, from the range 1…15;
• column "I" has 5 numbers from 16…30;
• column "N" has 4 numbers from 31…45, and the center square is designated free, here represented by the letter F;
• column "G" has 5 numbers from 46…60;
• column "O" has 5 numbers from 61…75.

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 "G-59". 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:

• any of the 5 rows, such as 7-24-33-59-69 or 13-28-F-50-71;
• any of the 5 columns, such as 55-48-50-59-49 or 32-44-F-33-40;
• either of the 2 diagonals, such as 9-21-F-59-74.

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:

• Although there are 75 numbers, only 24 of them appear on any card. Hence, about two-thirds of the time a player will not be able to mark a square when a number is called.
• Segregating the 75 numbers into 5 groups of 15, each identified by a letter, greatly speeds the search process (and the game) when a number is called. For example, if "I-29" is called, a player need look through merely the 5 numbers in the "I" column, not all 24 numbers on the card.

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 freely-chosen 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 37-cell 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 time-consuming 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 37-cell card will not be 1…36, but rather be 36 two-digit values from the following six ranges:

10…15
20…25
30…35
40…45
50…55
60…65

Figure 2B has four examples of 37-cell 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 2B1figure 2B2
figure 2B3figure 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 61-cell 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 2B1repeated 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 non-hexagonal 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
5-straight, 6-straight, 7-straight
two other rotations
7-wye
one other rotation
5-sharp, 7-sharp
five other rotations
5-blunt, 7-blunt
five other rotations
6-triangle, 9-triangle, 10-triangle
one other rotation
6-hexagon, 7-hexagon
no other rotations

Some players might opt to recognize the 9-triangle, but only as long as the center remains empty. In other words, a 10-triangle would not qualify as a 9-triangle. Similar remarks apply to the 6- and 7-hexagons. 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 5-straight. 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 5-straight.

figure 3B
player 1player 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 — 47-34-38-51-59 player 2 — 51-58-54-15-68

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 second-highest 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:

• 5-straight: prize \$1;
• 6-straight: prize \$2;
• 7-straight: prize \$4.

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 7-straight … … that player wins \$7: \$4 for the 7-straight, \$2 for a 6-straight, \$1 for a 5-straight. The game ends. If the first claim is for a 6-straight … … that player wins \$3: \$2 for the 6-straight, \$1 for a 5-straight. Play continues until someone claims a 7-straight and wins \$4. The game ends. If the first claim is for a 5-straight … … that player wins \$1. If the next claim is for a 7-straight … … that player wins \$6: \$4 for the 7-straight, \$2 for a 6-straight. The game ends. If the next claim is for a 6-straight … … that player wins \$2. Play continues until someone claims a 7-straight 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 7-straight has priority over anyone with a 6-straight; and 6-straight over 5-straight. 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 6-straight with 27-25-24-20-19-15. Bill has a 5-straight with 59-58-51-46-42. Anne has priority over Bill because she has the longest straight, even though Bill's 5-straight has higher numbers than either 5-straight residing within Anne's 6-straight. Anne wins \$3 and Bill wins \$0.

Section 3D. Another pattern that might be used is the skip. On the 37-cell 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 1-11-24-34, 10-18-26-33, and 15-20-25-29. Cells that a 4-skip can never reach are marked with an 'X'.

figure 3D1 — 37 cells

Figure 3D2 highlights some 3-skips, and 3D3 some 5-skips.

figure 3D2 — 19 cells figure 3D3 — 61 cells

Some players may choose to allow a skip of less than maximal length, such as a 4-skip on a 61-cell 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 37-cell card to 30 cells, and the 61-cell to 55.

Section 4. By way of example, here are some cards that are not regular hexagons. As for zoning, two-digit 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.

• 4B1 has one free cell in the center, as is customary. However, this leads to a zone size of 17. Such might not be regarded as convenient, but it follows from the principle that all zones be the same size.
• 4B2, with 3 free cells, has 7 zones of 7 cells, a more moderate partition. This is another case of how the manipulation of free cells can improve the zoning.

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 6-straight that uses neither free cell ("the hard way") has priority over a 6-straight that uses one of them, which in turn has priority over the 6-straight that uses both.

figure 4D — 50 cells
6 zones of 8 cells8 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 (63-81-88 etc) to the bottom row (65-62-83 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 non-cells 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 41-47-57-54-50-16.

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:

• They can shout "bingo" and win \$1, but forfeit the chance to win any further prize on that card in that game.
• They can remain quiet, and hope (with no guarantee) to win a larger prize by completing a double or triple ring before anyone else claims one.

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.

 exterior interior examples of maximal straights see figure 5A 4-straight 44-63-62-642 others none 6-straight 51-56-53-23-22-242 others 32-30-31-26-27-232 others 8-straight none 13-12-50-54-55-26-20-255 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 multiple-prize 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 9-triangle pattern. Enough cells are provided so that any hole can be surrounded by a 9-triangle pointed either up or down. Also, every cell can be a part of at least one 9-triangle. (Edge-to-edge 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
5-straight, 7-straight
five other rotations
6-hexagon
no other rotations
12-ring
one other rotation

two different 6-straights
two other rotations
9-triangle
one other rotation
12-star
no other rotations
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 12-ring.

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 5-straight 31-39-64-63-68 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 5-straight
two other rotations
7-star
no other rotations
4-straight, 6-straight
five other rotations
6-ring
one other rotation
 12-ringno other rotations double 6-ring, 9 cellstwo other rotations double 6-ring, 11 cellstwo 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: 60-95-90-93-94-91-25-22-23-24-63-65.

Besides the usual per-game 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 5-straight
two other rotations
major 8-ring
no other rotations
minor 5-cross
no other rotations

4-straight, 6-straight
five other rotations
minor 8-ring
no other rotations
major 5-cross
no other rotations

Section 10. Bingo can be played on a card that represents a three-dimensional shape, in this example a cube. Figure 10A below depicts such a card, numbered straight through for reference.

• Cells 1…16 can be visualized as the top layer of the cube;
• 17…32 immediately below them;
• 33…48 next;
• and 49…64 on the bottom.

figure 10A — 64 cells

Figure 10B is a typical card. While the entire card is a 4 × 4 × 4 cube, each zone happens to be a 2 × 2 × 2 cube. One zone for instance includes cells 30…37.

figure 10B — 64 cells
8 zones of 8 cells, 0 free cells

In the interest of simplicity, players might limit winning patterns to straights of length four. Using the cards above, here are some non-diagonal examples:

 horizontal vertical altitudinal figure 10A figure 10B figure 10A figure 10B figure 10A figure 10B 2-6-10-14 36-32-15-11 9-10-11-12 12-15-66-67 1-17-33-49 33-30-20-21 19-23-27-31 75-76-60-65 17-18-19-20 30-35-75-70 7-23-39-55 71-76-87-86 33-37-41-43 20-22-47-41 45-46-47-48 41-45-57-53 10-26-42-58 15-14-42-40 52-56-60-64 81-83-50-51 53-54-55-56 27-25-86-83 16-32-48-64 61-62-53-51 12 others 12 others 12 others

Here are some diagonals of one kind:

 ver-alt alt-hor hor-ver figure 10A figure 10B figure 10A figure 10B figure 10A figure 10B 5-22-39-56 31-37-87-83 4-24-44-64 74-77-55-51 1-6-11-16 33-32-66-67 16-31-46-61 61-65-45-46 14-26-38-50 11-14-24-23 45-42-39-36 41-42-87-80 6 others 6 others 6 others

and diagonals of another kind:

 hor-ver-alt figure 10A figure 10B 13-26-39-52 17-14-87-81 1-22-43-64 33-37-52-51 2 others

Some players might prefer the zoning arrangement of figure 10C, because each zone lies entirely within one layer:

figure 10C — 64 cells
8 zones of 8 cells, 0 free cells

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.