Python Puzzles
Sicherman dice are a pair of 6-sided dice with non-standard numbers–one with the sides 1, 2, 2, 3, 3, 4 and the other with the sides 1, 3, 4, 5, 6, 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)