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Card Flip Challenge

A deck of 90 cards is numbered from 1 to 90. Each card has a blue side and an orange side, with the number printed on both sides.

Alice arranges all the cards on a table with their blue sides facing up. She then follows these steps:

1. Flip over every card with a number that is a multiple of 3.
2. Flip over every card with a number that is a multiple of 4.
3. Flip over every card with a number that is a multiple of 5.

After Alice has completed all these steps, how many cards have their orange side facing up?

FLIP 🃏 The Card Flip Challenge Can you track the multiples and figure out the final colors? 📋 The 3 Rules Start: All 90 cards are BLUE side up. Step 1: Flip multiples of 3 (3, 6, 9...) Step 2: Flip multiples of 4 (4, 8, 12...) Step 3: Flip multiples of 5 (5, 10, 15...) 🔍 The Flipping Logic ? Start State ? 1 Flip (Odd) ? 2 Flips (Even) A card only ends up ORANGE if it flips an ODD number of times! Example Board: The First 15 Cards & Their Flip History 1- 2- 33 44 55 63 7- 84 93 105 11- 123, 4 13- 14- 153, 5 Challenge: After all steps, how many of the 90 cards are ORANGE?
✅ Solution: The Card Flip Challenge Using set theory and LCM (Least Common Multiple) to find the exact overlaps. 1️⃣ Count the Multiples Total Flips in each set (1 to 90): Multiples of 3: 90 ÷ 3 = 30 Multiples of 4: 90 ÷ 4 = 22.5 ➔ 22 Multiples of 5: 90 ÷ 5 = 18 Flipped Twice (Overlaps): 3 & 4 (LCM 12): 90 ÷ 12 = 7.5 ➔ 7 3 & 5 (LCM 15): 90 ÷ 15 = 6 4 & 5 (LCM 20): 90 ÷ 20 = 4.5 ➔ 4 Flipped Thrice (The Center): 3, 4, & 5 (LCM 60): 90 ÷ 60 = 1.5 ➔ 1 (Only the card '60' flips all 3 times!) Finding the "Only 1 Flip" cards: Only 3: 30 - 7 - 6 + 1 = 18 Only 4: 22 - 7 - 4 + 1 = 12 Only 5: 18 - 6 - 4 + 1 = 10 2️⃣ Map the Overlaps Odd Flips (Ends Orange) Even Flips (Ends Blue) Multiples of 3 Multiples of 4 Multiples of 5 18 Cards 12 Cards 10 Cards 6 Cards 5 Cards 3 Cards 1 Card 3️⃣ Add up the Orange Cards (1 Flip + 3 Flips) 18 + 12 + 10 + 1 = 41 Total Orange Cards Facing Up