Puzzle for November 6, 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
In eq.2, add A to each side, and subtract E from each side: E – C + A – E = B – A + A – E which becomes eq.2a) –C + A = B – E In eq.4, replace D + E with B – D – F (from eq.5): A + C = B – D – F + F which becomes eq.4a) A + C = B – D
Hint #2
Subtract the left and right sides of eq.2a from the left and right sides of eq.4a, respectively: A + C – (–C + A) = B – D – (B – E) which is equivalent to A + C + C – A = B – D – B + E which becomes 2×C = –D + E Add D to both sides of the above equation: 2×C + D = –D + E + D which becomes eq.4b) 2×C + D = E
Hint #3
In eq.6, replace E with 2×C + D (from eq.4b): 2×C + D + C = D – C which becomes 3×C + D = D – C In the equation above, subtract D from each side, and add C to each side: 3×C + D – D + C = D – C – D + C which makes 4×C = 0 which means C = 0
Hint #4
In eq.4b, substitute 0 for C: 2×0 + D = E which makes D = E
Hint #5
Substitute 0 for C, and D for E in eq.4: A + 0 = D + D + F which becomes eq.4c) A = 2×D + F
Hint #6
Substitute D for E, and 2×D + F for A (from eq.4c) in eq.3: B + D + D – F = 2×D + F + F which becomes B + 2×D – F = 2×D + 2×F In the equation above, subtract 2×D from each side, and add F to each side: B + 2×D – F – 2×D + F = 2×D + 2×F – 2×D + F which makes eq.3a) B = 3×F
Hint #7
Substitute D for E, and 3×F for B in eq.5: D + D = 3×F – D – F which becomes 2×D = 2×F – D Add D to both sides of the above equation: 2×D + D = 2×F – D + D which becomes 3×D = 2×F Divide both sides by 2: 3×D ÷ 2 = 2×F ÷ 2 which makes 1½×D = F
Hint #8
Substitute 1½×D for F in eq.4c: A = 2×D + 1½×D which makes A = 3½×D
Hint #9
Substitute (1½×D) for F in eq.3a: B = 3×(1½×D) which makes B = 4½×D
Solution
Substitute 3½×D for A, 4½×D for B, 0 for C, D for E, and 1½×D for F in eq.1: 3½×D + 4½×D + 0 + D + D + 1½×D = 23 which simplifies to 11½×D = 23 Divide both sides of the equation above by 11½: 11½×D ÷ 11½ = 23 ÷ 11½ which means D = 2 making A = 3½×D = 3½ × 2 = 7 B = 4½×D = 4½ × 2 = 9 D = E = 2 F = 1½×D = 1½ × 2 = 3 and ABCDEF = 790223