Puzzle for December 31, 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
In eq.3, subtract F from each side, and add D to both sides: C – A + F – F + D = A – D – F + D which becomes C – A + D = A – F In eq.2, replace A – F with C – A + D: D + B – A = C – A + D In the equation above, add A to both sides, and subtract D from both sides: D + B – A + A – D = C – A + D + A – D which simplifies to B = C
Hint #2
In eq.6, replace B with C: F = C + C which makes F = 2×C
Hint #3
In eq.4, replace F with 2×C, and replace B with C: D – 2×C = C – D Add D and 2×C to both sides of the above equation: D – 2×C + D + 2×C = C – D + D + 2×C 2×D = 3×C Divide both sides by 2: 2×D ÷ 2 = 3×C ÷ 2 which makes D = 1½×C
Hint #4
In eq.5, substitute 2×C for F, C for B, and 1½×C for D: E + 2×C = C + 1½×C which becomes E + 2×C = 2½×C Subtract 2×C from both sides of the equation above: E + 2×C – 2×C = 2½×C – 2×C which makes E = ½×C
Hint #5
In eq.2, substitute C for B, 1½×C for D, and 2×C for F: C + 1½×C – A = A – 2×C which becomes 2½×C – A = A – 2×C Add A and 2×C to each side of the above equation: 2½×C – A + A + 2×C = A – 2×C + A + 2×C which becomes 4½×C = 2×A Divide both sides by 2: 4½×C ÷ 2 = 2×A ÷ 2 which makes eq.2a) 2¼×C = A
Solution
Substitute 2¼×C for A, C for B, 1½×C for D, ½×C for E, and 2×C for F in eq.1: 2¼×C + C + C + 1½×C + ½×C + 2×C = 33 which simplifies to 8¼×C = 33 Divide both sides of the above equation by 8¼: 8¼×C ÷ 8¼ = 33 ÷ 8¼ which means C = 4 making A = 2¼×C = 2¼ × 4 = 9 B = C = 4 D = 1½×C = 1½ × 4 = 6 E = ½×C = ½ × 4 = 2 F = 2×C = 2 × 4 = 8 and ABCDEF = 944628