Puzzle for August 13, 2022 ( )
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 B and D to both sides of eq.2: D – B + B + D = A – D + B + D which becomes eq.2a) 2×D = A + B Add D and F to both sides of eq.4: E – D + D + F = D – F + D + F which becomes eq.4a) E + F = 2×D
Hint #2
In eq.1, replace A + B with 2×D (from eq.2a), and E + F with 2×D (from eq.4a): 2×D + C + D + 2×D = 34 which becomes C + 5×D = 34 Subtract 5×D from each side of the equation above: C + 5×D – 5×D = 34 – 5×D which becomes eq.1a) C = 34 – 5×D
Hint #3
In eq.5, add F to both sides, and subtract D from both sides: C – F + F – D = B + D + F – D which becomes eq.5a) C – D = B + F
Hint #4
To make eq.1a true, check several possible values for D and C: If D = 4, then C = 34 – 5×4 = 34 – 20 = 14 If D = 5, then C = 34 – 5×5 = 34 – 25 = 9 If D = 6, then C = 34 – 5×6 = 34 – 30 = 4 If D = 7, then C = 34 – 5×7 = 34 – 35 = –1 If D > 7, then C < –1 If D < 4, then C > 14 Since C must be a one-digit non-negative integer, then the above equations make: D = 5 and C = 9 or D = 6 and C = 4
Hint #5
Check: D = 6, and C = 4 ... Substituting 4 for C, and 6 for D in eq.5a would yield: 4 – 6 = B + F which would make –2 = B + F Since B and F must be non-negative, then: –2 ≠ B + F which means D ≠ 6 and C ≠ 4 and therefore makes D = 5 and C = 9
Hint #6
Substitute 9 for C in eq.3: B – E = F – 9 Add E and 9 to both sides of the above equation: B – E + E + 9 = F – 9 + E + 9 which becomes B + 9 = F + E which is the same as eq.3a) B + 9 = E + F
Hint #7
Substitute B + 9 for E + F (from eq.3a), and 5 for D in eq.4a: B + 9 = 2×5 which becomes B + 9 = 10 Subtract 9 from each side of the above equation: B + 9 – 9 = 10 – 9 which makes B = 1
Hint #8
Substitute 5 for D, and 1 for B in eq.2a: 2×5 = A + 1 which becomes 10 = A + 1 Subtract 1 from both sides of the above equation: 10 – 1 = A + 1 – 1 which makes 9 = A
Hint #9
Substitute 9 for C, 5 for D, and 1 for B in eq.5a: 9 – 5 = 1 + F which becomes 4 = 1 + F Subtract 1 from both sides of the equation above: 4 – 1 = 1 + F – 1 which makes 3 = F
Solution
Substitute 3 for F, and 5 for D in eq.4a: E + 3 = 2×5 which becomes E + 3 = 10 Subtract 3 from each side of the equation above: E + 3 – 3 = 10 – 3 which makes E = 7 and makes ABCDEF = 919573