The wizard has three magical coins. When cast into the air, they reveal their true nature. Let X be the number of mystical tails that appear. Discover the possible values of X and the ancient runes (inverse images) that summon them.
Step 1: The three coins can reveal these arcane patterns:
H
H
H
H
H
T
H
T
H
H
T
T
T
H
H
T
H
T
T
T
H
T
T
T
Step 2: Count the mystical tails in each pattern:
Arcane Pattern
Tails Count (X)
HHH
0
HHT, HTH, THH
1
HTT, THT, TTH
2
TTT
3
Step 3: The prophecy reveals X can be: {0, 1, 2, 3}
Step 4: Ancient runes that summon each value:
X⁻¹(0) = {HHH} → 1 rune
X⁻¹(1) = {HHT, HTH, THH} → 3 runes
X⁻¹(2) = {HTT, THT, TTH} → 3 runes
X⁻¹(3) = {TTT} → 1 rune
2
Enchanted Fruit Basket
The fairy queen's basket contains 5 golden mangoes and 4 ruby apples. Three fruits are taken by the forest sprites. If the number of apples taken is the random variable X, find all possible values of X and the magical combinations that produce them.
Step 1: The basket contains:
M
M
M
M
M
A
A
A
A
Step 2: Possible apple counts when drawing 3 fruits:
Step 3: X can be: {0, 1, 2, 3}
Step 4: Magical combinations:
Apples (X)
Fruit Combinations
Number of Ways
0
MMM (all mangoes)
C(5,3) = 10
1
MM + A
C(5,2) × C(4,1) = 10 × 4 = 40
2
M + AA
C(5,1) × C(4,2) = 5 × 6 = 30
3
AAA (all apples)
C(4,3) = 4
3
Dragon's Treasure Game
The dragon's urn contains 6 fire rubies (red) and 8 night pearls (black). You draw 2 gems. For each ruby you win ₹15, but lose ₹10 for each pearl. If X is your total winnings, find all possible values of X and the dragon's deals that lead to them.
Step 1: The dragon's treasure contains:
R
R
R
R
R
R
B
B
B
B
B
B
B
B
Step 2: Possible draws and winnings:
Combination
Calculation
Winnings (X)
Number of Ways
2 Rubies
15 + 15
+30
C(6,2) = 15
1 Ruby + 1 Pearl
15 - 10
+5
C(6,1) × C(8,1) = 48
2 Pearls
-10 - 10
-20
C(8,2) = 28
Step 3: Possible winnings: {-20, 5, 30}
4
Ancient Die of Destiny
A mystical die has one face marked '2', two faces marked '3', and three faces marked '4'. When thrown twice, X represents the total score. Decipher all possible values of X and the fateful rolls that produce them.