Puzzle for August 30, 2022 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
In eq.6, replace E with B + F (from eq.4): B + F – F = F – B which becomes B = F – B Add B to both sides of the above equation: B + B = F – B + B which makes 2×B = F
Hint #2
In eq.4, replace F with 2×B: E = B + 2×B which makes E = 3×B
Hint #3
In eq.2, replace F with 2×B: 2×B = B + C Subtract B from each side of the above equation: 2×B – B = B + C – B which makes B = C
Hint #4
In eq.3, substitute B for C: eq.3a) A = B + D
Hint #5
In eq.5, substitute B + D for A (from eq.3a), and 3×B for E: B + D + D = 3×B – D which becomes B + 2×D = 3×B – D In the equation above, subtract B from both sides, and add D to both sides: B + 2×D – B + D = 3×B – D – B + D which makes 3×D = 2×B Divide both sides by 3: 3×D ÷ 3 = 2×B ÷ 3 which makes D = ⅔×B
Hint #6
Substitute ⅔×B for D in eq.3a: A = B + ⅔×B which makes A = 1⅔×B
Solution
Substitute 1⅔×B for A, B for C, ⅔×B for D, 3×B for E, and 2×B for F in eq.1: 1⅔×B + B + B + ⅔×B + 3×B + 2×B = 28 which simplifies to 9⅓×B = 28 Divide both sides of the above equation by 9⅓: 9⅓×B ÷ 9⅓ = 28 ÷ 9⅓ which means B = 3 making A = 1⅔×B = 1⅔ × 3 = 5 C = B = 3 D = ⅔×B = ⅔ × 3 = 2 E = 3×B = 3 × 3 = 9 F = 2×B = 2 × 3 = 6 and ABCDEF = 533296