As you probably know, a bear levels up when you receive a duplicate of it.

After a particular bear reaches its maximum level (for most types of bears this occurs at Level 100), further duplicates of the same bear will not make it more powerful. Rather, these duplicates will reduce the nap time of the bear. These nap time reductions have diminishing returns. I.e., the first few times it happens, the nap time will decrease by a lot. However, later on the nap time will decrease by very little.

- To be more exact, nap times approach 40% of their original value as you get more and more duplicates of a particular bear beyond the maximum level. The approach is quick at first, but gets extremely slow toward the end.
- When you click a bear on the bear selection screen, we display the nap time in the largest sensible unit of measure and don't display fractional parts. E.g., 3.89 hours is displayed as "3 hours". This is just what gets displayed; the actual nap time is still 3.89 hours.

## An Example

Skinny Bear in version 01.03.00 has a max level of 100 and a base nap time of 60 minutes.

Number of Bears | Excess Bears | Fraction by which naptime is reduced from the base | Fraction of base naptime remaining | Skinny Bear's actual naptime (in minutes) | Skinny Bear's displayed naptime (in minutes except where noted) |

1-100 | 0 | 0 | 1 | 60 | 1 hour |

101 | 1 | 0.15 | 0.85 | 51 | 51 |

102 | 2 | 0.24 | 0.76 | 45.6 | 45 |

103 | 3 | 0.3 | 0.7 | 42 | 42 |

104 | 4 | 0.342857 | 0.657143 | 39.4386 | 39 |

105 | 5 | 0.375 | 0.625 | 37.5 | 37 |

106 | 6 | 0.4 | 0.6 | 36 | 36 |

107 | 7 | 0.42 | 0.58 | 34.8 | 34 |

108 | 8 | 0.436364 | 0.563636 | 33.8181 | 33 |

109 | 9 | 0.45 | 0.55 | 33 | 33 |

110 | 10 | 0.461538 | 0.538462 | 32.3070 | 32 |

111 | 11 | 0.471429 | 0.528571 | 31.7143 | 31 |

112 | 12 | 0.48 | 0.52 | 31.2 | 31 |

113 | 13 | 0.4875 | 0.5125 | 30.75 | 30 |

114 | 14 | 0.494118 | 0.505882 | 30.3529 | 30 |

115 | 15 | 0.5 | 0.5 | 30 | 30 |

116 | 16 | 0.505263 | 0.494737 | 29.6842 | 29 |

117 | 17 | 0.51 | 0.49 | 29.4 | 29 |

118 | 18 | 0.514286 | 0.485714 | 29.1429 | 29 |

119 | 19 | 0.518182 | 0.481818 | 28.9091 | 28 |

120 | 20 | 0.521739 | 0.478261 | 28.6957 | 28 |

121 | 21 | 0.525 | 0.475 | 28.5 | 28 |

122 | 22 | 0.528 | 0.472 | 28.32 | 28 |

123 | 23 | 0.530769 | 0.469231 | 28.1538 | 28 |

124 | 24 | 0.533333 | 0.466667 | 28 | 28 |

125 | 25 | 0.535714 | 0.464286 | 27.8571 | 27 |

126 | 26 | 0.537931 | 0.462069 | 27.7241 | 27 |

127 | 27 | 0.54 | 0.46 | 27.6 | 27 |

128 | 28 | 0.541935 | 0.458065 | 27.4839 | 27 |

129 | 29 | 0.54375 | 0.45625 | 27.375 | 27 |

130 | 30 | 0.545454 | 0.454546 | 27.2727 | 27 |

. . . | |||||

Infinity | Infinity | 0.6 | 0.4 | 24 | 24 |

Note the following:

- Going from your 100th Skinny Bear to your 101st Skinny Bear reduces the nap time by 9 minutes.
- Going from your 101st Skinny Bear to your 102nd Skinny Bear reduces the nap time by 5.4 minutes. (I.e., the returns are diminishing.)
- At your 115th bear, the nap time is 50% of its base value.
- A bear can never go below 40% of its base nap time value.
- When the nap time isn't an exact number of minutes, we don't display fractional part but the actual nap time still includes the fractional part. (This is more of an issue when the nap time is displayed in hours because we won't display the minutes even though there may be so many minutes that it's almost closer to an additional hour.)

## The Math-y Details

The fraction by which naptime is reduced is given by this formula:

**Nap Time Reduction Fraction = 0.6 * (Excess Bears) / (Excess Bears + 3)**

where Excess Bears is the number of bears you've earned beyond the Max Level. E.g., if you have acquired 103 Skinny Bears, then Excess Bears is 3 because Skinny Bear's Max Level is 100.

This gives the following exact points of reference:

- Nap time is reduced by 30% if you have 3 Excess Bears.
- Nap time is reduced by 40% if you have 6 Excess Bears.
- Nap time is reduced by 50% if you have 15 Excess Bears.
- Nap time is reduced by 60% if you have infinity Excess Bears.

Or, written another way:

- Naptime is 70% of normal if you have 3 Excess Bears.
- Naptime is 60% of normal if you have 6 Excess Bears.
- Naptime is 50% of normal if you have 15 Excess Bears.
- Naptime is 40% of normal if you have infinity Excess Bears.

If you're in between those exact points, you'll have to do the math to figure out exactly what the nap time should be.

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