Puzzle for June 24, 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 D and E to both sides of eq.4: C - D + D + E = D - E + D + E which becomes eq.4a) C + E = 2×D
Hint #2
eq.6 may be written as: A + B = C + E + D In the above equation, replace C + E with 2×D (from eq.4a): A + B = 2×D + D which becomes eq.6a) A + B = 3×D
Hint #3
Add C to both sides of eq.3: E + C = A + D + C which may be written as eq.3a) C + E = A + D + C
Hint #4
In eq.3a, replace C + E with 2×D (from eq.4a): 2×D = A + D + C Subtract D from each side of the equation above: 2×D - D = A + D + C - D which becomes eq.3b) D = A + C
Hint #5
In eq.3b, substitute C + F for D (from eq.2): C + F = A + C Subtract C from each side of the above equation: C + F - C = A + C - C which makes F = A
Hint #6
Add A and E to both sides of eq.5: B + C + A + E = E + F + A + E which becomes B + C + A + E = 2×E + F + A which may be written as eq.5a) A + B + C + E = 2×E + F + A
Hint #7
Substitute 3×D for A + B (from eq.6a), 2×D for C + E (from eq.4a), and A for F in eq.5a: 3×D + 2×D = 2×E + A + A which becomes eq.5b) 5×D = 2×E + 2×A
Hint #8
Substitute (A + D) for E (from eq.3) into eq.5b: 5×D = 2×(A + D) + 2×A which becomes 5×D = 2×A + 2×D + 2×A which becomes 5×D = 4×A + 2×D Subtract 2×D from both sides of the equation above: 5×D - 2×D = 4×A + 2×D - 2×D which makes 3×D = 4×A Divide both sides by 4: 3×D ÷ 4 = 4×A ÷ 4 which makes ¾×D = A and also makes ¾×D = A = F
Hint #9
Substitute ¾×D for A in eq.3: E = ¾×D + D which makes E = 1¾×D
Hint #10
Substitute 1¾×D for E in eq.4a: C + 1¾×D = 2×D Subtract 1¾×D from each side of the equation above: C + 1¾×D - 1¾×D = 2×D - 1¾×D which makes C = ¼×D
Hint #11
Substitute ¾×D for A in eq.6a: ¾×D + B = 3×D Subtract ¾×D from each side of the above equation: ¾×D + B - ¾×D = 3×D - ¾×D which makes B = 2¼×D
Solution
Substitute ¾×D for A and F, 2¼×D for B, ¼×D for C, and 1¾×D for E in eq.1: ¾×D + 2¼×D + ¼×D + D + 1¾×D + ¾×D = 27 which simplifies to 6¾×D = 27 Divide both sides of the above equation by 6¾: 6¾×D ÷ 6¾ = 27 ÷ 6¾ which means D = 4 making A = F = ¾×D = ¾ × 4 = 3 B = 2¼×D = 2¼ × 4 = 9 C = ¼×D = ¼ × 4 = 1 E = 1¾×D = 1¾ × 4 = 7 and ABCDEF = 391473