Puzzle for May 29, 2019 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit positive integer.
Scratchpad
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Hint #1
In eq.2, replace E with B + D (from eq.3): D + B + D = A which becomes eq.2a) B + 2×D = A
Hint #2
Add C to both sides of eq.5: A - C + C = C - F + C which becomes A = 2×C - F In the above equation, substitute (D + F) for C (from eq.4): A = 2×(D + F) - F which is equivalent to A = 2×D + 2×F - F which becomes eq.5a) A = 2×D + F
Hint #3
In eq.5a, substitute B + 2×D for A (from eq.2a): B + 2×D = 2×D + F Subtract 2×D from each side of the above equation: B + 2×D - 2×D = 2×D + F - 2×D which makes B = F
Hint #4
In eq.4, substitute B for F: C = D + B which may be written as C = B + D Substitute C for B + D in eq.3: E = C
Hint #5
Substitute B for F in eq.6: B = D × B Divide each side of the above equation by B: B ÷ B = D × B ÷ B which means 1 = D
Hint #6
Substitute 1 for D in eq.2: eq.2b) 1 + E = A
Hint #7
Substitute 1 for D in eq.3: E = B + 1 Subtract 1 from both sides: E - 1 = B + 1 - 1 which makes E - 1 = B and which also makes eq.3a) F = B = E - 1
Solution
Substitute 1 + E for A (from eq.2b), E - 1 for B and F (from eq.3a), E for C, and 1 for D in eq.1: 1 + E + E - 1 + E + 1 + E + E - 1 = 30 which simplifies to 5×E = 30 Divide both sides of the above equation by 5: 5×E ÷ 5 = 30 ÷ 5 which means E = 6 making A = 1 + E = 1 + 6 = 7 (from eq.2b) B = F = E - 1 = 6 - 1 = 5 (from eq.3a) C = E = 6 and ABCDDEF = 756165