Puzzle for February 1, 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 B + C with F (from eq.2): D + F = F + E Subtract F from each side of the above equation: D + F – F = F + E – F which makes D = E
Hint #2
In eq.4, replace F with A + D (from eq.3): D + E = A + A + D which becomes D + E = 2×A + D Subtract D from each side of the equation above: D + E – D = 2×A + D – D which makes E = 2×A and also makes D = E = 2×A
Hint #3
In eq.3, substitute 2×A for D: F = A + 2×A which makes F = 3×A
Hint #4
Substitute 3×A for F, and 2×A for E in eq.5: B + 3×A = C + 2×A – B Subtract 2×A from both sides of the above equation: B + 3×A – 2×A = C + 2×A – B – 2×A which becomes eq.5a) 2×B + A = C
Hint #5
Substitute 3×A for F, and 2×B + A for C (from eq.5a) in eq.2: 3×A = B + 2×B + A which becomes 3×A = 3×B + A Subtract A from each side of the equation above: 3×A – A = 3×B + A – A which makes 2×A = 3×B Divide both sides by 3: 2×A ÷ 3 = 3×B ÷ 3 which makes ⅔×A = B
Hint #6
Substitute (⅔×A) for B in eq.5a: 2×(⅔×A) + A = C which becomes 1⅓×A + A = C which makes 2⅓×A = C
Solution
Substitute ⅔×A for B, 2⅓×A for C, 2×A for D and E, and 3×A for F in eq.1: A + ⅔×A + 2⅓×A + 2×A + 2×A + 3×A = 33 which simplifies to 11×A = 33 Divide both sides of the above equation by 11: 11×A ÷ 11 = 33 ÷ 11 which means A = 3 making B = ⅔×A = ⅔ × 3 = 2 C = 2⅓×A = 2⅓ × 3 = 7 D = E = 2×A = 2 × 3 = 6 F = 3×A = 3 × 3 = 9 and ABCDEF = 327669