Python Puzzles
Sicherman dice are a pair of 6-sided dice with non-standard numbers–one with the sides 1, 2, 2, 3, 3, and 4, and the other with the sides 1, 3, 4, 5, 6, and 8. They are notable as the only pair of 6-sided dice that are not normal dice, bear only positive integers, and have the same probability distribution for the sum as normal dice.
Run a simulation to prove Sicherman dice have the same probability as a normal pair of dice.
Observing the results, it is evident that the counts are remarkably similar.
ID Dice Sum Normal Sicherman Difference
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0 2 28,159 27,650 509
1 3 55,499 55,425 74
2 4 83,553 83,034 519
3 5 111,067 111,101 -34
4 6 139,156 139,617 -461
5 7 166,948 166,435 513
6 8 138,000 138,616 -616
7 9 110,680 111,222 -542
8 10 83,369 83,920 -551
9 11 55,872 55,414 458
10 12 27,697 27,566 131import numpy as np
import pandas as pd
# Define dice
standard_dice = np.array([1, 2, 3, 4, 5, 6])
sicherman_dice_1 = np.array([1, 2, 2, 3, 3, 4])
sicherman_dice_2 = np.array([1, 3, 4, 5, 6, 8])
# Define number of simulations
num_simulations = 1000000
# Simulate dice throws
standard_sums = np.random.choice(standard_dice, num_simulations) + np.random.choice(standard_dice, num_simulations)
sicherman_sums = np.random.choice(sicherman_dice_1, num_simulations) + np.random.choice(sicherman_dice_2, num_simulations)
# Create dataframes
standard_df = pd.DataFrame(standard_sums, columns=['DiceSum'])
standard_df['Type'] = 'Standard'
sicherman_df = pd.DataFrame(sicherman_sums, columns=['DiceSum'])
sicherman_df['Type'] = 'Sicherman'
# Concatenate dataframes
df = pd.concat([standard_df, sicherman_df])
# Get counts
counts_df = df.groupby(['Type', 'DiceSum']).size().reset_index(name='Counts')
# Pivot the dataframe
pivot_df = counts_df.pivot(index='DiceSum', columns='Type', values='Counts').reset_index()
# Compute the difference
pivot_df['Difference'] = pivot_df['Sicherman'] - pivot_df['Standard']
# Rename columns
pivot_df.columns = ['DiceSum', 'Normal Count', 'Sicherman Count', 'Difference']
# Format numbers with commas
pivot_df['Normal Count'] = pivot_df['Normal Count'].apply('{:,}'.format)
pivot_df['Sicherman Count'] = pivot_df['Sicherman Count'].apply('{:,}'.format)
pivot_df['Difference'] = pivot_df['Difference'].apply('{:,}'.format)
print(pivot_df)