Puzzle for June 30, 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
Add F to both sides of eq.4: A + C + E + F = B + D - F + F which becomes eq.4a) A + C + E + F = B + D eq.1 may be written as: eq.1a) B + D + A + C + E + F = 36
Hint #2
In eq.1a, replace A + C + E + F with B + D (from eq.4a): B + D + B + D = 36 which may be written as 2×(B + D) = 36 Divide both sides of the above equation by 2: 2×(B + D) ÷ 2 = 36 ÷ 2 which becomes eq.1b) B + D = 18
Hint #3
Since B and D are one-digit non-negative integers, then: 0 ≤ B ≤ 9 and 0 ≤ D ≤ 9 making 0 ≤ B + D ≤ 18 Combining all the above inequalities with eq.1b makes: B = 9 and D = 9
Hint #4
In eq.2, substitute 9 for B: eq.2a) 9 = A + F
Hint #5
eq.4a may be written as: A + F + C + E = B + D Substitute 9 for A + F (from eq.2a), and 18 for B + D (from eq.1b) in the above equation: 9 + C + E = 18 Subtract 9 and C from each side: 9 + C + E - 9 - C = 18 - 9 - C which becomes eq.4b) E = 9 - C
Hint #6
Substitute 9 for D, and 9 - C for E (from eq.4b) in eq.3: 9 + C = 9 - C + F - C which becomes 9 + C = 9 - 2×C + F In the equation above, subtract 9 from both sides, and add 2×C to both sides: 9 + C - 9 + 2×C = 9 - 2×C + F - 9 + 2×C which simplifies to 3×C = F
Hint #7
Substitute 3×C for F in eq.5: 3×C ÷ C = A - 3×C which becomes 3 = A - 3×C Add 3×C to both sides of the above equation: 3 + 3×C = A - 3×C + 3×C which becomes eq.5a) 3 + 3×C = A
Hint #8
Substitute 3 + 3×C for A (from eq.5a), 9 - C for E (from eq.4b), 3×C for F, and 18 for B + D (from eq.1b) in eq.4a: 3 + 3×C + C + 9 - C + 3×C = 18 which becomes 12 + 6×C = 18 Subtract 12 from each side of the above equation: 12 + 6×C - 12 = 18 - 12 which becomes 6×C = 6 Divide both sides by 6: 6×C ÷ 6 = 6 ÷ 6 which makes C = 1
Solution
Since C = 1, then: A = 3 + 3×C = 3 + (3×1) = 3 + 3 = 6 (from eq.5a) E = 9 - C = 9 - 1 = 8 (from eq.4b) F = 3×C = 3×1 = 3 and ABCDEF = 691983