Puzzle for March 24, 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 non-negative integer.
* CD is a 2-digit number (not C×D).
Scratchpad
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Hint #1
Add B + C to both sides of eq.3: F - B + B + C = A - C + B + C which becomes F + C = A + B In the above equation, replace A + B with D (from eq.2): F + C = D which may be written as eq.3a) C + F = D
Hint #2
Add the left and right sides of eq.4 to the left and right sides of eq.5, respectively: A + C + B + F = B + E + F + E - D which may be written as eq.5a) (A + B) + (C + F) = (B + F) + 2×E - D
Hint #3
In eq.5a, substitute D for A + B (from eq.2), D for C + F (from eq.3a), and (E - D) for B + F (from eq.4): D + D = (E - D) + 2×E - D which becomes 2×D = 3×E - 2×D Add 2×D to both sides of the equation above: 2×D + 2×D = 3×E - 2×D + 2×D which becomes 4×D = 3×E Divide both sides by 4: 4×D ÷ 4 = 3×E ÷ 4 which becomes D = ¾×E
Hint #4
eq.1 may be written as: A + B + C + F + D + E = 26 In the equation above, substitute D for A + B (from eq.2), and D for C + F (from eq.3a): D + D + D + E = 26 which becomes 3×D + E = 26 Substitute (¾×E) for D: 3×(¾×E) + E = 26 which is equivalent to 2¼×E + E = 26 which means 3¼×E = 26 Divide both sides by 3¼: 3¼×E ÷ 3¼ = 26 ÷ 3¼ which makes E = 8 and which makes D = ¾×E = ¾ × 8 = 6
Hint #5
eq.3 may be written as: A - C = F - B Add the left and right sides of the above equation to the left and right sides of eq.5, respectively: A + C + A - C = B + E + F + F - B which becomes 2×A = E + 2×F Substitute 8 for E in the above equation: 2×A = 8 + 2×F Divide both sides by 2: 2×A ÷ 2 = (8 + 2×F) ÷ 2 which becomes A = 4 + F Subtract 4 from both sides: A - 4 = 4 + F - 4 which makes eq.5a) A - 4 = F
Hint #6
In eq.6, substitute D for A + B: CD = (D + F) × E which may be written as 10×C + D = (D + F) × E Substitute 6 for D, and 8 for E: 10×C + 6 = (6 + F) × 8 which becomes 10×C + 6 = 48 + 8×F Subtract 6 from each side: 10×C + 6 - 6 = 48 + 8×F - 6 which becomes eq.6a) 10×C = 42 + 8×F
Hint #7
Substitute (A - 4) for F (from eq.5a) in eq.6a: 10×C = 42 + 8×(A - 4) which is equivalent to 10×C = 42 + 8×A - 32 which becomes 10×C = 10 + 8×A Divide both sides by 10: 10×C ÷ 10 = (10 + 8×A) ÷ 10 which becomes eq.6b) C = 1 + ⅘×A
Hint #8
Substitute 1 + ⅘×A for C (from eq.6b), A - 4 for F (from eq.5a), and 6 for D in eq.3a: 1 + ⅘×A + A - 4 = 6 which becomes 1⅘×A - 3 = 6 Add 3 to each side: 1⅘×A - 3 + 3 = 6 + 3 which becomes 1⅘×A = 9 Divide both sides by 1⅘: 1⅘×A ÷ 1⅘ = 9 ÷ 1⅘ which makes A = 5
Hint #9
Substitute 5 for A in eq.5a: 5 - 4 = F which makes 1 = F
Hint #10
Substitute 5 for A in eq.6b: C = 1 + ⅘×5 which makes C = 1 + 4 = 5
Solution
Substitute 5 for A, and 6 for D in eq.2: 6 = 5 + B Subtract 5 from both sides: 6 - 5 = 5 + B - 5 which means 1 = B making ABCDEF = 515681