Puzzle for April 27, 2023 ( )
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
Subtract C and B from both sides of eq.4: C + F - C - B = A + B - C - B which becomes F - B = A - C Add the left and right sides of the above equation to the left and right sides of eq.3, respectively: B + E + F - B = A + C + A - C which becomes eq.3a) E + F = 2×A
Hint #2
In eq.5, replace E + F with 2×A (from eq.3a): 2×A - A = A - E which becomes A = A - E Subtract A from each side of the equation above: A - A = A - E - A which makes 0 = -E which means 0 = E
Hint #3
In eq.3a, substitute 0 for E: 0 + F = 2×A which makes F = 2×A
Hint #4
In eq.2, substitute 2×A for F, and 0 for E: D = 2×A - 0 which makes D = 2×A
Hint #5
In eq.3, substitute 0 for E: B + 0 = A + C which becomes eq.3b) B = A + C
Hint #6
eq.6 may be written as: C = (B + D + E + F) ÷ 4 Multiply both sides of the above equation by 4: 4 × C = 4 × (B + D + E + F) ÷ 4 which becomes eq.6a) 4×C = B + D + E + F
Hint #7
Substitute A + C for B (from eq.3b), 2×A for D and F, and 0 for E in eq.6a: 4×C = A + C + 2×A + 0 + 2×A which becomes 4×C = 5×A + C Subtract C from each side of the above equation: 4×C - C = 5×A + C - C which makes 3×C = 5×A Divide both sides by 3: 3×C ÷ 3 = 5×A ÷ 3 which makes C = 1⅔×A
Hint #8
Substitute 1⅔×A for C in eq.3b: B = A + 1⅔×A which makes B = 2⅔×A
Solution
Substitute 2⅔×A for B, 1⅔×A for C, 2×A for D and F, and 0 for E in eq.1: A + 2⅔×A + 1⅔×A + 2×A + 0 + 2×A = 28 which simplifies to 9⅓×A = 28 Divide both sides of the above equation by 9⅓: 9⅓×A ÷ 9⅓ = 28 ÷ 9⅓ which means A = 3 making B = 2⅔×A = 2⅔ × 3 = 8 C = 1⅔×A = 1⅔ × 3 = 5 D = F = 2×A = 2 × 3 = 6 and ABCDEF = 385606