Puzzle for September 15, 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.
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Hint #1
In eq.4, substitute (A + B) for D (from eq.2): C - (A + B) = E + F - C which becomes C - A - B = E + F - C Add A, B, and C to both sides of the above equation: C - A - B + A + B + C = E + F - C + A + B + C which becomes eq.4a) 2×C = E + F + A + B
Hint #2
eq.1 may be written as: E + F + A + B + C + D = 35 In the above equation, replace E + F + A + B with 2×C (from eq.4a): 2×C + C + D = 35 which becomes 3×C + D = 35 Subtract 3×C from each side: 3×C + D - 3×C = 35 - 3×C which becomes eq.1a) D = 35 - 3×C
Hint #3
To make eq.1a true, check several possible values for C and D: If C = 9, then D = 35 - 3×9 = 35 - 27 = 8 If C = 8, then D = 35 - 3×8 = 35 - 24 = 11 If C < 8, then D > 11 Since C and D must be one-digit non-negative integers, the above equations make: C = 9 and D = 8
Hint #4
In eq.2, replace D with 8: 8 = A + B Subtract A from each side of the equation above: 8 - A = A + B - A which becomes eq.2a) 8 - A = B
Hint #5
In eq.5, substitute 8 - A for B (from eq.2a), 8 for D, and 9 for C: 8 - A + 8 - A = A + 9 - 8 which becomes 16 - 2×A = A + 1 In the above equation, add 2×A to both sides, and subtract 1 from both sides: 16 - 2×A + 2×A - 1 = A + 1 + 2×A - 1 which makes 15 = 3×A Divide both sides by 3: 15 ÷ 3 = 3×A ÷ 3 which makes 5 = A
Hint #6
Substitute 5 for A in eq.2a: 8 - 5 = B which makes 3 = B
Hint #7
Substitute 5 for A in eq.3: F - E = E - 5 Add E to both sides of the equation above: F - E + E = E - 5 + E which becomes eq.3a) F = 2×E - 5
Hint #8
Substitute 9 for C, 2×E - 5 for F (from eq.3a), 5 for A, and 3 for B in eq.4a: 2×9 = E + 2×E - 5 + 5 + 3 which becomes 18 = 3×E + 3 Subtract 3 from each side of the above equation: 18 - 3 = 3×E + 3 - 3 which makes 15 = 3×E Divide both sides by 3: 15 ÷ 3 = 3×E ÷ 3 which makes 5 = E
Solution
Substitute 5 for E in eq.3a: F = 2×5 - 5 which becomes F = 10 - 5 which makes F = 5 and makes ABCDEF = 539855