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Last Updated: March 16, 2021

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Stairs

Introduction

Stairs is a game found at BetFury Internet casino. It involves randomly placing ladders to climb levels. Randomly on each level are placed rocks. If the player places his ladder at a rock, he loses. This is shown as the rock falling on the player's head. Otherwise, his balance grows (except in one situation). After each successful pick, the player may cash in or advance further.

Players should be warned that contrary to the playing field, the player may not place his ladder in every position. In other words, some ladder locations are blocked off.

stairs

Rules

  1. The game has 13 levels.
  2. The player may choose to 1 to 7 rocks to be randomly placed on each level.

     

  3. The player will start on what I will call level 0.
  4. The player must choose a position to place his ladder to climb up to level 1. There are 20 such positions to place the ladder.
  5. If the player places his ladder at a position with a hidden rock, then he will lose. Otherwise, he will successfully move on.
  6. Contrary to what the game looks like, the number of positions decreases by one with each level. The following table shows what appear to be the number of positions per level and the number of actual positions.

     

    Positions by Level

    Level Visible
    Positions
    Actual
    Positions
    1 20 20
    2 19 19
    3 19 18
    4 17 17
    5 19 16
    6 15 15
    7 17 14
    8 13 13
    9 12 12
    10 19 11
    11 10 10
    12 9 9
    13 8 8

     

  7. If the player tries to place his ladder at a spot where there is no true position, then a slash through a circle icon will indicate the choice is not allowed. If the player chooses auto-pick, the game will never choose an invalid position.
  8. After the player successfully rises a level, he may either cash out his win or take his chances and climb another level.
  9. The table below shows the win (on a "for one" basis) for attaining each level according to the number of rocks on each level.

 

Wins by Level and Number of rocks

Level 1 rock 2 rocks 3 rocks 4 rocks 5 rocks 6 rocks 7 rocks
1 1.00 1.06 1.12 1.19 1.27 1.36 1.46
2 1.06 1.18 1.33 1.50 1.72 1.98 2.31
3 1.12 1.33 1.59 1.93 2.38 2.98 3.79
4 1.19 1.50 1.93 2.53 3.37 4.60 6.44
5 1.27 1.72 2.38 3.37 4.90 7.36 11.44
6 1.36 1.98 2.98 4.60 7.36 12.26 21.46
7 1.46 2.31 3.79 6.44 11.44 21.46 42.92
8 1.58 2.73 4.92 9.30 18.60 39.85 92.98
9 1.73 3.28 6.56 13.95 31.88 79.70 223.16
10 1.90 4.01 9.03 21.92 58.45 175.34 613.70
11 2.11 5.01 12.89 36.53 116.90 438.36 2045.67
12 2.38 6.45 19.34 65.75 263.01 1315.07 9205.50
13 2.71 8.60 30.94 131.51 701.37 5260.29 73644.00

Analysis

The following table shows the win (on a for one basis), probability of winning, and expected return for all combinations of number of rocks and level. As you can see, the expected return never strays far from 95%.

 

Stairs Analysis

rocks Level Pays Prob. Win Exp. Ret.
1 1 1 0.950000 0.950000
1 2 1.06 0.900000 0.954000
1 3 1.12 0.850000 0.952000
1 4 1.19 0.800000 0.952000
1 5 1.27 0.750000 0.952500
1 6 1.36 0.700000 0.952000
1 7 1.46 0.650000 0.949000
1 8 1.58 0.600000 0.948000
1 9 1.73 0.550000 0.951500
1 10 1.9 0.500000 0.950000
1 11 2.11 0.450000 0.949500
1 12 2.38 0.400000 0.952000
1 13 2.71 0.350000 0.948500
2 1 1.06 0.900000 0.954000
2 2 1.18 0.805263 0.950211
2 3 1.33 0.715789 0.952000
2 4 1.5 0.631579 0.947368
2 5 1.72 0.552632 0.950526
2 6 1.98 0.478947 0.948316
2 7 2.31 0.410526 0.948316
2 8 2.73 0.347368 0.948316
2 9 3.28 0.289474 0.949474
2 10 4.01 0.236842 0.949737
2 11 5.01 0.189474 0.949263
2 12 6.45 0.147368 0.950526
2 13 8.6 0.110526 0.950526
3 1 1.12 0.850000 0.952000
3 2 1.33 0.715789 0.952000
3 3 1.59 0.596491 0.948421
3 4 1.93 0.491228 0.948070
3 5 2.38 0.399123 0.949912
3 6 2.98 0.319298 0.951509
3 7 3.79 0.250877 0.950825
3 8 4.92 0.192982 0.949474
3 9 6.56 0.144737 0.949474
3 10 9.03 0.105263 0.950526
3 11 12.89 0.073684 0.949789
3 12 19.34 0.049123 0.950035
3 13 30.94 0.030702 0.949912
4 1 1.19 0.800000 0.952000
4 2 1.5 0.631579 0.947368
4 3 1.93 0.491228 0.948070
4 4 2.53 0.375645 0.950382
4 5 3.37 0.281734 0.949443
4 6 4.6 0.206605 0.950382
4 7 6.44 0.147575 0.950382
4 8 9.3 0.102167 0.950155
4 9 13.95 0.068111 0.950155
4 10 21.92 0.043344 0.950093
4 11 36.53 0.026006 0.950006
4 12 65.75 0.014448 0.949948
4 13 131.51 0.007224 0.950021
5 1 1.19 0.750000 0.892500
5 2 1.5 0.552632 0.828947
5 3 1.93 0.399123 0.770307
5 4 2.53 0.281734 0.712786
5 5 3.37 0.193692 0.652742
5 6 4.6 0.129128 0.593989
5 7 6.44 0.083011 0.534590
5 8 9.3 0.051084 0.475077
5 9 13.95 0.029799 0.415693
5 10 21.92 0.016254 0.356285
5 11 36.53 0.008127 0.296877
5 12 65.75 0.003612 0.237487
5 13 131.51 0.001354 0.178129
6 1 1.36 0.700000 0.952000
6 2 1.98 0.478947 0.948316
6 3 2.98 0.319298 0.951509
6 4 4.6 0.206605 0.950382
6 5 7.36 0.129128 0.950382
6 6 12.26 0.077477 0.949865
6 7 21.46 0.044272 0.950087
6 8 39.85 0.023839 0.949985
6 9 79.7 0.011920 0.949985
6 10 175.34 0.005418 0.949985
6 11 438.36 0.002167 0.950006
6 12 1315.07 0.000722 0.949999
6 13 5260.29 0.000181 0.950001
7 1 1.46 0.650000 0.949000
7 2 2.31 0.410526 0.948316
7 3 3.79 0.250877 0.950825
7 4 6.44 0.147575 0.950382
7 5 11.44 0.083011 0.949644
7 6 21.46 0.044272 0.950087
7 7 42.92 0.022136 0.950087
7 8 92.98 0.010217 0.949950
7 9 223.16 0.004257 0.949985
7 10 613.7 0.001548 0.950000
7 11 2045.67 0.000464 0.950002
7 12 9205.5 0.000103 0.950000
7 13 73644 0.000013 0.950000

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