Puzzle for November 19, 2020 ( )
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
eq.5 may be written as: A + E = D + C + F In the equation above, replace C + F with D + E (from eq.3): A + E = D + D + E which becomes A + E = 2×D + E Subtract E from each side: A + E – E = 2×D + E – E which makes A = 2×D
Hint #2
In eq.4, replace B with A + F (from eq.6): E + F = A + F + D Subtract F from each side of the equation above: E + F – F = A + F + D – F which becomes eq.4a) E = A + D
Hint #3
In eq.4a, substitute 2×D for A: E = 2×D + D which makes E = 3×D
Hint #4
Substitute 2×D for A in eq.2: C + D = 2×D + B Subtract D from both sides of the equation above: C + D – D = 2×D + B – D which becomes C = D + B which may be written as eq.2a) C = B + D
Hint #5
Substitute C for B + D in eq.4: eq.4b) E + F = C
Hint #6
Substitute E + F for C (from eq.4b) in eq.3: D + E = E + F + F which becomes D + E = E + 2×F Subtract E from both sides of the above equation: D + E – E = E + 2×F – E which makes D = 2×F Divide both sides by 2: D ÷ 2 = 2×F ÷ 2 which makes ½×D = F
Hint #7
Substitute 3×D for E, and ½×D for F in eq.4b: 3×D + ½×D = C which makes 3½×D = C
Hint #8
Substitute 2×D for A, and ½×D for F in eq.6: B = 2×D + ½×D which makes B = 2½×D
Solution
Substitute 2×D for A, 2½×D for B, 3½×D for C, 3×D for E, and ½×D for F in eq.1: 2×D + 2½×D + 3½×D + D + 3×D + ½×D = 25 which simplifies to 12½×D = 25 Divide both sides of the above equation by 12½: 12½×D ÷ 12½ = 25 ÷ 12½ which means D = 2 making A = 2×D = 2 × 2 = 4 B = 2½×D = 2½ × 2 = 5 C = 3½×D = 3½ × 2 = 7 E = 3×D = 3 × 2 = 6 F = ½×D = ½ × 2 = 1 and ABCDEF = 457261