Puzzle for August 28, 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.
Scratchpad
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Hint #1
eq.5 may be written as: A + B + F = C + E + D In the above equation, replace C + E with B + D + F (from eq.6): A + B + F = B + D + F + D which becomes A + B + F = B + 2×D + F Subtract both B and F from each side of the equation above: A + B + F – B – F = B + 2×D + F – B – F which simplifies to A = 2×D
Hint #2
In eq.2, replace A with 2×D: D + F = 2×D Subtract D from both sides of the equation above: D + F – D = 2×D – D which makes F = D
Hint #3
Add A and E to both sides of eq.4: C – A + A + E = D – E + A + E which becomes C + E = D + A In eq.6, substitute D + A for C + E: B + D + F = D + A Subtract D and F from both sides of the above equation: B + D + F – D – F = D + A – D – F which becomes eq.6a) B = A – F
Hint #4
Substitute 2×D for A, and D for F in eq.6a: B = 2×D – D which makes B = D
Hint #5
Substitute D for B and F in eq.3: E + D = D + C Subtract D from both sides of the equation above: E + D – D = D + C – D which makes E = C
Hint #6
Substitute D for B and F, and C for E in eq.6: D + D + D = C + C which means 3×D = 2×C Divide both sides by 2: 3×D ÷ 2 = 2×C ÷ 2 which makes 1½×D = C and also makes E = C = 1½×D
Solution
Substitute 2×D for A, D for B and F, and 1½×D for C and E in eq.1: 2×D + D + 1½×D + D + 1½×D + D = 16 which simplifies to 8×D = 16 Divide both sides of the equation above by 8: 8×D ÷ 8 = 16 ÷ 8 which means D = 2 making A = 2×D = 2 × 2 = 4 B = F = D = 2 C = E = 1½×D = 1½ × 2 = 3 and ABCDEF = 423232