Puzzle for January 4, 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
Add C and F to both sides of eq.3: E – F + C + F = A – C + C + F which becomes eq.3a) E + C = A + F eq.4 may be written as: B + C = A + F + D In the above equation, replace A + F with E + C (from eq.3a): B + C = E + C + D Subtract C from both sides: B + C – C = E + C + D – C which becomes eq.4a) B = E + D
Hint #2
Add C and D to both sides of eq.5: A – D + C + D = E – C + C + D which becomes eq.5a) A + C = E + D In eq.4a, replace E + D with A + C (from eq.5a): eq.4b) B = A + C
Hint #3
eq.1 may be written as: A + C + B + E + D + F = 33 In the equation above, substitute B for A + C (from eq.4b) and for E + D (from eq.4a): B + B + B + F = 33 which becomes 3×B + F = 33 Subtract 3×B from both sides of the above equation: 3×B + F – 3×B = 33 – 3×B which becomes eq.1a) F = 33 – 3×B
Hint #4
To make eq.1a true, check some possible one-digit non-negative integers for B and F: If B = 9, then F = 33 – 3×9 = 33 – 27 = 6 If B = 8, then F = 33 – 3×8 = 33 – 24 = 9 If B = 7, then F = 33 – 3×7 = 33 – 21 = 12 If B < 7, then F > 12 The only one-digit non-negative integers for B and F that make eq.1a true are: Case 1: B = 9, and F = 6 or Case 2: B = 8, and F = 9
Hint #5
Check Case 1: B = 9, and F = 6 ... Substituting 6 for F, and 9 for B in eq.2 would yield: 6 = A + 9 Subtracting 9 from both sides of eq.2 would yield: 6 – 9 = A + 9 – 9 which would make –3 = A Since A must be non-negative, then B ≠ 9 and F ≠ 6 making B = 8 and F = 9
Hint #6
Substitute 9 for F, and 8 for B in eq.2: 9 = A + 8 Subtract 8 from each side of the above equation: 9 – 8 = A + 8 – 8 which makes 1 = A
Hint #7
Substitute 8 for B, and 1 for A in eq.4b: 8 = 1 + C Subtract 1 from both sides of the equation above: 8 – 1 = 1 + C – 1 which means 7 = C
Hint #8
Substitute 9 for F, 1 for A, and 7 for C in eq.3: E – 9 = 1 – 7 which becomes E – 9 = –6 Add 9 to both sides of the above equation: E – 9 + 9 = –6 + 9 which makes E = 3
Solution
Substitute 1 for A, 3 for E, and 7 for C in eq.5: 1 – D = 3 – 7 which becomes 1 – D = –4 Add both D and 4 to each side of the above equation: 1 – D + D + 4 = –4 + D + 4 which means 5 = D making ABCDEF = 187539