Puzzle for November 30, 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, replace A + B with E + F (from eq.2): E + F + C = D + E Subtract E from both sides of the above equation: E + F + C – E = D + E – E which becomes eq.3a) F + C = D
Hint #2
In eq.5, add C to both sides, and subtract D from both sides: A – C + C – D = D + F + C – D which becomes A – D = F + C In the above equation, replace F + C with D (from eq.3a): A – D = D Add D to both sides: A – D + D = D + D which makes A = 2×D
Hint #3
Add the left and right sides of eq.2 to the left and right sides of eq.4, respectively: A – B + (A + B) = D – F + (E + F) which becomes 2×A = D + E Substitute (2×D) for A in the equation above: 2×(2×D) = D + E which becomes 4×D = D + E Subtract D from both sides: 4×D – D = D + E – D which makes 3×D = E
Hint #4
Substitute 2×D for A, and 3×D for E in eq.3: 2×D + B + C = D + 3×D Subtract 2×D from each side of the equation above: 2×D + B + C – 2×D = D + 3×D – 2×D which becomes eq.3b) B + C = 2×D
Hint #5
Substitute 2×D for A, and 3×D for E in eq.2: 2×D + B = 3×D + F Subtract 2×D from both sides of the above equation: 2×D + B – 2×D = 3×D + F – 2×D which becomes eq.2a) B = D + F
Hint #6
Substitute 2×D for both A and B + C (from eq.3b), and 3×D for E in eq.1: 2×D + 2×D + D + 3×D + F = 25 which becomes 8×D + F = 25 Subtract 8×D from each side of the above equation: 8×D + F – 8×D = 25 – 8×D which makes eq.1a) F = 25 – 8×D
Hint #7
To make eq.1a true, check several possible values for D and F: If D = 0 then F = 25 – 8×0 = 25 – 0 = 25 If D = 1 then F = 25 – 8×1 = 25 – 8 = 17 If D = 2 then F = 25 – 8×2 = 25 – 16 = 9 If D = 3 then F = 25 – 8×3 = 25 – 24 = 1 If D = 4 then F = 25 – 8×4 = 25 – 32 = –7 If D > 5 then F < –7 Since D and F must be one-digit non-negative integers, then either: D = 2 and F = 9 or: D = 3 and F = 1
Hint #8
Check: D = 2, and F = 9 ... Substituting 2 for D, and 9 for F in eq.2a would yield: B = 2 + 9 which would make B = 11 Since B must be a one-digit integer, then: B ≠ 11 which means D ≠ 2 and F ≠ 9 making D = 3 and F = 1 and also making A = 2×D = 2 × 3 = 6 E = 3×D = 3 × 3 = 9
Hint #9
Substitute 3 for D, and 1 for F in eq.2a: B = 3 + 1 which makes B = 4
Solution
Substitute 1 for F, and 3 for D in eq.3a: 1 + C = 3 Subtract 1 from both sides of the equation above: 1 + C – 1 = 3 – 1 which makes C = 2 making ABCDEF = 642391