Puzzle for December 28, 2021 ( )
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.3, replace A with E + F (from eq.2): D + F = E + F + E which becomes D + F = 2×E + F Subtract F from both sides of the equation above: D + F – F = 2×E + F – F which makes eq.3a) D = 2×E
Hint #2
In eq.5, replace B + F with A + D (from eq.4): C + D = A + A + D which becomes C + D = 2×A + D Subtract D from each side of the equation above: C + D – D = 2×A + D – D which makes C = 2×A
Hint #3
In eq.4, substitute E + F for A (from eq.2), and 2×E for D: B + F = E + F + 2×E which becomes B + F = 3×E + F Subtract F from each side of the above equation: B + F – F = 3×E + F – F which makes eq.4a) B = 3×E
Hint #4
Substitute 2×A for C, and 3×E for B in eq.6: A + 2×A = 3×E + E which makes 3×A = 4×E Divide both sides of the above equation by 4: 3×A ÷ 4 = 4×E ÷ 4 which makes ¾×A = E
Hint #5
Substitute (¾×A) for E in eq.4a: B = 3×(¾×A) which makes B = 2¼×A
Hint #6
Substitute (¾×A) for E in eq.3a: D = 2×(¾×A) which makes D = 1½×A
Hint #7
Substitute ¾×A for E in eq.2: ¾×A + F = A Subtract ¾×A from each side of the equation above: ¾×A + F – ¾×A = A – ¾×A which makes F = ¼×A
Solution
Substitute 2¼×A for B, 2×A for C, 1½×A for D, ¾×A for E, and ¼×A for F in eq.1: A + 2¼×A + 2×A + 1½×A + ¾×A + ¼×A = 31 which simplifies to 7¾×A = 31 Divide both sides of the above equation by 7¾: 7¾×A ÷ 7¾ = 31 ÷ 7¾ which means A = 4 making B = 2¼×A = 2¼ × 4 = 9 C = 2×A = 2 × 4 = 8 D = 1½×A = 1½ × 4 = 6 E = ¾×A = ¾ × 4 = 3 F = ¼×A = ¼ × 4 = 1 and ABCDEF = 498631