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# Da Vinci Diamonds Keno

## Introduction

Davinci Diamonds is a keno variant. I first noticed it in 2021 at both VideoPoker.com and Resorts World in Las Vegas. In addition to the usual spot keno rules, the game places there 2x2 paintings on the field of numbers. The player can win multipliers and a bonus round if enough tiles of a painting are covered. In the bonus round, the player gets free spins in which catching numbers on the paintings wins extra balls.

## Rules

- After making a bet, the player picks 2 to 10 numbers from the range of 1 to 80. The player may also choose to have this step done randomly.
- The player will select three 2x2 regions on the field of numbers that do not include any numbers already picked. These squares will be marked with four-panel paintings, one number per panel. The player may also choose to let the game choose these regions randomly.
- The game shall draw 20 numbers, without replacement, from the range of 1 to 80.
- If a number drawn by the game matches a number chosen by the player, then it shall be known as a "catch."
- If a number drawn by the game matches one covered by a painting, then that panel of the painting shall be illuminated.
- The player shall be paid based on how many numbers he chose and the number of catches.
- If exactly three of the four panels of a painting are illuminated, then the player shall win a 2x multiplier.
- If the player wins more than one multiplier, the multipliers are added.
- Any multiplier won is in lieu of the 1x multiplier he gets for not earning any multipliers.
- If the player covers all four panels of any painting, then he shall play the free spin bonus game.
- Free spins shall be played with the same number of picks and bet amount as the initiating spin.
- The number of initial free spins in the bonus game shall be the product of the number of illuminated tiles among the three paintings and the number of paintings entirely illuminated. For example, the maximum free initial free spins would be 12×3 = 36.
- Free spins shall play like initial spins, except if a ball drawn matches a number in a painting, then the player shall win two extra balls. These extra balls are represented as gems, a different color gem for each painting.
- If an extra ball matches a number in a painting, the player shall win two more balls. This process shall continue until no additional numbers drawn match a number covered by a painting.
- Only after all balls are drawn is the win calculated for a free spin.
- Free spins can trigger more free spins, up to a total of 600 free spins.

If the rules above were unclear, below are the rule screens from the help files. Click on any image for a larger version.

## Video

The following video explains and demonstrates how to play Da Vinci Diamonds keno.

## Examples

The image above is from an initial game. With no multiplier, the win for catching 6 is 6. Note the painting in the upper left corner has been 3/4 covered, as evidenced that three of the four panels are illuminated. This earns the player a 2x multiplier. Thus, his win for that game is 2 × 6 = 12.

The image above is from a bonus game, immediately after the 20-ball draw. The player has already five catches. Note also that between the three paintings, five tiles have been hit. The player is entitled to two extra balls, which are illustrated as gems, per tile, so he gets 2 × 5 = 10 more balls.

The image above is from the same bonus game, after the 10 additional balls. Two of them were additional catches, on numbers 46 and 47, bringing the number of catches from five to seven. Two of the extra balls hit panels on paintings, on numbers 6 and 32. Thus, the player is entitled to 2 × 2 = 4 extra balls.

The image above is from the same bonus game. The four extra balls his 21, 27, 41, 42, none of which hit anything, ending that bonus game. Note the upper painting, covering 5, 6, 15, and 16 was hit three times, on 5, 6, and 16. Hitting that painting three times earned the player a 2x multiplier. Normally catching seven numbers pays 33. With the 2x multiplier, the players win that game is 2 × 33 = 66.

## Analysis

### Base Game Probabilities

Catch | Pick 2 | Pick 3 | Pick 4 | Pick 5 | Pick 6 | Pick 7 | Pick 8 | Pick 9 | Pick 10 |
---|---|---|---|---|---|---|---|---|---|

0 | 0.5601266 | 0.4165044 | 0.3083214 | 0.2271842 | 0.1666018 | 0.1215743 | 0.0882662 | 0.0637478 | 0.0457907 |

1 | 0.3797468 | 0.4308666 | 0.4327318 | 0.4056861 | 0.3634947 | 0.3151925 | 0.2664641 | 0.2206656 | 0.1795714 |

2 | 0.0601266 | 0.1387537 | 0.2126355 | 0.2704574 | 0.3083214 | 0.3266541 | 0.3281456 | 0.3164261 | 0.2952568 |

3 | 0.0138754 | 0.0432479 | 0.0839351 | 0.1298195 | 0.1749932 | 0.2147862 | 0.2461092 | 0.2674024 | |

4 | 0.0030634 | 0.0120923 | 0.0285379 | 0.0521910 | 0.0815037 | 0.1141052 | 0.1473189 | ||

5 | 0.0006449 | 0.0030956 | 0.0086385 | 0.0183026 | 0.0326015 | 0.0514277 | |||

6 | 0.0001290 | 0.0007321 | 0.0023667 | 0.0057196 | 0.0114794 | ||||

7 | 0.0000244 | 0.0001605 | 0.0005917 | 0.0016111 | |||||

8 | 0.0000043 | 0.0000326 | 0.0001354 | ||||||

9 | 0.0000007 | 0.0000061 | |||||||

10 | 0.0000001 | ||||||||

Total | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

In a regular game, it is easy to see the number of painting tiles is 20 × (12/80) = 3. The table below shows the probability of all specific number of tiles hit, from 0 to 12.

### Number of Tiles Hit

Tiles Hit | Probability |
---|---|

0 | 0.0232271671 |

1 | 0.1137657162 |

2 | 0.2377703470 |

3 | 0.2797298199 |

4 | 0.2057628002 |

5 | 0.0993873148 |

6 | 0.0322088520 |

7 | 0.0070273859 |

8 | 0.0010195984 |

9 | 0.0000954010 |

10 | 0.0000054280 |

11 | 0.0000001673 |

12 | 0.0000000021 |

Total | 1.0000000000 |

After a lot of math, the following shows the probability of each multiplier.

- 1x multiplier (0 paintings 3/4 covered) = 0.8742857942
- 2x multiplier (1 painting 3/4 covered) = 0.1217123190
- 4x multiplier (2 paintings 3/4 covered) = 0.0039741316
- 6x multiplier (3 paintings 3/4 covered) = 0.0000277551

The average multiplier on an initial game is 1.133773489. Note three is a negative correlation between the number of catches and multiplier size.

The following shows the probability of each possible number of paintings entirely covered in an initial game.

- 0 paintings 4/4 covered = 0.9908228580
- 1 painting 4/4 covered = 0.0091641092
- 2 paintings 4/4 covered = 0.0000130307
- 3 paintings 4/4 covered = 0.0000000021

The probability of one or more paintings entirely covered, thus earning this bonus is 0.009177142, or about 1 in 109. The average number of paintings covered is 0.009190177.

### Pick 2

The following table shows the probability and contribution for the return in the base game for the pick-2 game with the 1-5 pay table.

### Pick 2 Return Table -- 1,5 Pay Table

Win | Probability | Return |
---|---|---|

0 | 0.560123 | 0.000000 |

1 | 0.335377 | 0.335377 |

2 | 0.043109 | 0.086218 |

4 | 0.001268 | 0.005072 |

5 | 0.054070 | 0.270349 |

6 | 0.000008 | 0.000046 |

10 | 0.005903 | 0.059025 |

20 | 0.000142 | 0.002843 |

30 | 0.000001 | 0.000020 |

1.000000 | 0.758949 |

The bonus probability is 0.009177. The mean, median and mode of the bonus win are 19.809981, 11, and 3 respectively. The contribution to the return from the bonus is 0.181823. The total return is 0.940772.

### Pick 3

The following table shows the probability and contribution for the return in the base game for the pick-3 game with the 2-25 pay table.

### Pick 3 Return Table -- 2,25 Pay Table

Win | Probability | Return |
---|---|---|

0 | 0.847367 | 0.000000 |

2 | 0.124316 | 0.248631 |

4 | 0.014090 | 0.056362 |

8 | 0.000354 | 0.002831 |

12 | 0.000002 | 0.000020 |

25 | 0.012641 | 0.316019 |

50 | 0.001206 | 0.060299 |

100 | 0.000024 | 0.002436 |

150 | 0.000000 | 0.000012 |

Total | 1.000000 | 0.686611 |

The bonus probability is 0.009177. The mean, median and mode of the bonus win are 29.350781, 8, and 2 respectively. The contribution to the return from the bonus is 0.269290 The total return is 0.955901.

### Pick 4

The following table shows the probability and contribution for the return in the base game for the pick-4 game with the 1-5-55 pay table.

### Pick 4 Return Table -- 1,5,55 Pay Table

Win | Probability | Return |
---|---|---|

0 | 0.741062 | 0.000000 |

1 | 0.189697 | 0.189697 |

2 | 0.022350 | 0.044699 |

4 | 0.000584 | 0.002336 |

5 | 0.039272 | 0.196361 |

6 | 0.000003 | 0.000018 |

10 | 0.003888 | 0.038875 |

20 | 0.000082 | 0.001647 |

30 | 0.000000 | 0.000010 |

55 | 0.002825 | 0.155359 |

110 | 0.000233 | 0.025614 |

220 | 0.000004 | 0.000864 |

330 | 0.000000 | 0.000005 |

Total | 1.000000 | 0.655484 |

The bonus probability is 0.009177. The mean, median and mode of the bonus win are 31.939606, 10, and 2 respectively. The contribution to the return from the bonus is 0.292905. The total return is 0.948389.

### Pick 5

The following table shows the probability and contribution for the return in the base game for the pick-5 game with the 2,17,214 pay table.

### Pick 5 Return Table -- 2,17,214 Pay Table

Win | Probability | Return |
---|---|---|

0 | 0.903317 | 0.000000 |

2 | 0.075941 | 0.151882 |

4 | 0.007828 | 0.031312 |

8 | 0.000172 | 0.001374 |

12 | 0.000001 | 0.000008 |

17 | 0.011128 | 0.189170 |

34 | 0.000952 | 0.032368 |

68 | 0.000017 | 0.001139 |

102 | 0.000000 | 0.000006 |

214 | 0.000603 | 0.128937 |

428 | 0.000042 | 0.018045 |

856 | 0.000001 | 0.000483 |

Total | 1.000000 | 0.554725 |

The bonus probability is 0.009177. The mean, median and mode of the bonus win are 43.138236, 8, and 0 respectively. The contribution to the return from the bonus is 0.395898. The total return is 0.950623.

### Pick 6

The following table shows the probability and contribution for the return in the base game for the pick-6 game with the 1-5-55-355 pay table.

### Pick 6 Return Table -- 1-5-55-355 Pay Table

Win | Probability | Return |
---|---|---|

0 | 0.838426 | 0.000000 |

1 | 0.116989 | 0.116989 |

2 | 0.012535 | 0.025071 |

4 | 0.000288 | 0.001152 |

5 | 0.026163 | 0.130815 |

6 | 0.000001 | 0.000007 |

10 | 0.002332 | 0.023316 |

20 | 0.000042 | 0.000850 |

30 | 0.000000 | 0.000004 |

55 | 0.002881 | 0.158447 |

110 | 0.000211 | 0.023162 |

220 | 0.000003 | 0.000660 |

330 | 0.000000 | 0.000003 |

355 | 0.000122 | 0.043166 |

710 | 0.000007 | 0.005108 |

1420 | 0.000000 | 0.000103 |

2130 | 0.000000 | 0.000001 |

Total | 1.000000 | 0.528853 |

The bonus probability is 0.009177. The mean, median and mode of the bonus win are 45.872830, 8, and 1 respectively. The contribution to the return from the bonus is 0.421108. The total return is 0.949961.

### Pick 7

The following table shows the probability and contribution for the return in the base game for the pick-7 game with the 1-3-10-105-500 pay table.

### Pick 7 Return Table -- 1,3,10,105,500 Pay Table

Win | Probability | Return |
---|---|---|

0 | 0.763410 | 0.000000 |

1 | 0.157062 | 0.157062 |

2 | 0.017522 | 0.035045 |

3 | 0.047680 | 0.143041 |

4 | 0.000420 | 0.001680 |

6 | 0.004427 | 0.026561 |

10 | 0.008018 | 0.080177 |

12 | 0.000084 | 0.001014 |

18 | 0.000000 | 0.000005 |

20 | 0.000609 | 0.012187 |

40 | 0.000009 | 0.000357 |

60 | 0.000000 | 0.000001 |

105 | 0.000690 | 0.072445 |

210 | 0.000042 | 0.008877 |

420 | 0.000000 | 0.000209 |

500 | 0.000023 | 0.011632 |

630 | 0.000000 | 0.000001 |

1000 | 0.000001 | 0.001149 |

2000 | 0.000000 | 0.000015 |

3000 | 0.000000 | 0.000001 |

Total | 1.000000 | 0.551460 |

The bonus probability is 0.009177. The mean, median and mode of the bonus win are 43.187140, 12, and 3 respectively. The contribution to the return from the bonus is 0.396457. The total return is 0.947917.

### Pick 8

The following table shows the probability and contribution for the return in the base game for the pick-8 game with the 2-7-53-150-1000 pay table.

### Pick 7 Return Table -- 2-7-53-150-1000 Pay Table

Win | Probability | Return |
---|---|---|

0 | 0.897675 | 0.000000 |

2 | 0.074192 | 0.148385 |

4 | 0.007163 | 0.028652 |

7 | 0.016934 | 0.118535 |

8 | 0.000143 | 0.001142 |

12 | 0.000000 | 0.000006 |

14 | 0.001342 | 0.018787 |

28 | 0.000021 | 0.000580 |

42 | 0.000000 | 0.000002 |

53 | 0.002221 | 0.117713 |

106 | 0.000142 | 0.015086 |

150 | 0.000153 | 0.023007 |

212 | 0.000002 | 0.000357 |

300 | 0.000008 | 0.002317 |

318 | 0.000000 | 0.000001 |

600 | 0.000000 | 0.000041 |

1000 | 0.000004 | 0.004136 |

2000 | 0.000000 | 0.000340 |

Total | 1.000000 | 0.479087 |

The bonus probability is 0.009177. The mean, median and mode of the bonus win are 51.244819, 11, and 0 respectively. The contribution to the return from the bonus is 0.470327. The total return is 0.949414.

### Pick 9

The following table shows the probability and contribution for the return in the base game for the pick-9 game with the 2,4,15,60,215,1500 pay table.

### Pick 9 Return Table -- 2,4,15,60,215,1500 Pay Table

Win | Probability | Return |
---|---|---|

0 | 0.846945 | 0.000000 |

2 | 0.103474 | 0.206949 |

4 | 0.040495 | 0.161978 |

8 | 0.002700 | 0.021599 |

12 | 0.000001 | 0.000010 |

15 | 0.005357 | 0.080351 |

16 | 0.000040 | 0.000643 |

24 | 0.000000 | 0.000002 |

30 | 0.000358 | 0.010743 |

60 | 0.000567 | 0.034017 |

90 | 0.000000 | 0.000001 |

120 | 0.000030 | 0.003572 |

215 | 0.000031 | 0.006729 |

240 | 0.000000 | 0.000062 |

360 | 0.000000 | 0.000000 |

430 | 0.000001 | 0.000545 |

860 | 0.000000 | 0.000008 |

1500 | 0.000001 | 0.001048 |

3000 | 0.000000 | 0.000057 |

Total | 1.000000 | 0.528313 |

The bonus probability is 0.009177. The mean, median and mode of the bonus win are 45.951903, 16, and 4 respectively. The contribution to the return from the bonus is 0.421687. The total return is 0.950000.

### Pick 10

The following table shows the probability and contribution for the return in the base game for the pick-10 game with the 1,3,6,33,155,700,2000 pay table.

### Pick 10 Return Table -- 1,3,6,33,155,700,2000 Pay Table

Win | Probability | Return |
---|---|---|

0 | 0.788025 | 0.000000 |

1 | 0.133052 | 0.133052 |

2 | 0.013968 | 0.027936 |

3 | 0.047279 | 0.141837 |

4 | 0.000303 | 0.001214 |

6 | 0.014797 | 0.088783 |

12 | 0.000815 | 0.009779 |

18 | 0.000000 | 0.000004 |

24 | 0.000010 | 0.000233 |

33 | 0.001524 | 0.050281 |

36 | 0.000000 | 0.000001 |

66 | 0.000084 | 0.005567 |

132 | 0.000001 | 0.000104 |

155 | 0.000130 | 0.020114 |

310 | 0.000006 | 0.001730 |

620 | 0.000000 | 0.000025 |

700 | 0.000006 | 0.004149 |

1400 | 0.000000 | 0.000268 |

2000 | 0.000000 | 0.000218 |

2800 | 0.000000 | 0.000003 |

4000 | 0.000000 | 0.000010 |

Total | 1.000000 | 0.485308 |

The bonus probability is 0.009177. The mean, median and mode of the bonus win are 50.662541, 14, and 6 respectively. The contribution to the return from the bonus is 0.464855. The total return is 0.950163.

## Pay Table 1

The following table shows the summary of my analysis of Pay Table 1, which is used at VideoPoker.com.

### Pay Table 1 Summary

Pick | Pay Table |
Base Game Return |
Bonus Return |
Total Return |
---|---|---|---|---|

2 | 1,5 | 0.758949 | 0.181823 | 0.940772 |

3 | 2,25 | 0.686611 | 0.269290 | 0.955901 |

4 | 1,5,55 | 0.655484 | 0.292905 | 0.948389 |

5 | 2,17,214 | 0.554725 | 0.395898 | 0.950623 |

6 | 1,5,55,355 | 0.528853 | 0.421108 | 0.949961 |

7 | 1,3,10,105,500 | 0.551460 | 0.396457 | 0.947917 |

8 | 2,7,53,150,1000 | 0.479087 | 0.470327 | 0.949414 |

9 | 2,4,15,60,215,1500 | 0.528313 | 0.421687 | 0.950000 |

10 | 1,3,6,33,155,700,2000 | 0.485308 | 0.464855 | 0.950163 |

## Pay Table 2

The following table shows the summary of my analysis of Pay Table 2, which I found at Resorts World in Las Vegas.

### Pay Table 2 Summary

Pick | Pay Table |
Base Game Return |
Bonus Return |
Total Return |
---|---|---|---|---|

2 | 1,4 | 0.692510 | 0.160660 | 0.853170 |

3 | 2,22 | 0.641077 | 0.246713 | 0.887790 |

4 | 1,5,47 | 0.629208 | 0.272261 | 0.901469 |

5 | 2,16,200 | 0.531757 | 0.375358 | 0.907115 |

6 | 1,5,49,345 | 0.507625 | 0.397850 | 0.905475 |

7 | 1,3,10,85,500 | 0.535883 | 0.368442 | 0.904325 |

8 | 2,7,48,132,1000 | 0.463607 | 0.442172 | 0.905779 |

9 | 2,4,13,52,215,1500 | 0.511122 | 0.394995 | 0.906117 |

10 | 1,3,6,27,138,700,2000 | 0.472821 | 0.432862 | 0.905683 |