Puzzle for January 12, 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, replace D with A + B (from eq.3): C + A + B = B + E + F Subtract B from each side of the above equation: C + A + B – B = B + E + F – B which becomes eq.2a) C + A = E + F
Hint #2
Subtract the left and right sides of eq.5 from the left and right sides of eq.2a, respectively: C + A – (A + E) = E + F – (C + F) which becomes C + A – A – E = E + F – C – F which becomes C – E = E – C Add C and E to both sides of the equation above: C – E + C + E = E – C + C + E which makes 2×C = 2×E Divide both sides by 2: 2×C ÷ 2 = 2×E ÷ 2 which means C = E
Hint #3
In eq.5, replace E with C: A + C = C + F Subtract C from both sides of the above equation: A + C – C = C + F – C which makes A = F
Hint #4
In eq.2, substitute C for E: C + D = B + C + F Subtract C from both sides of the equation above: C + D – C = B + C + F – C which becomes eq.2b) D = B + F
Hint #5
Substitute B + F for D (from eq.2b), and F for A in eq.4: B + F – C = F + F – B In the above equation, subtract F from each side, and add B to each side: B + F – C – F + B = F + F – B – F + B which makes 2×B – C = F which also makes eq.4a) A = F = 2×B – C
Hint #6
Substitute 2×B – C for F (from eq.4a) in eq.2b: D = B + 2×B – C which becomes eq.2c) D = 3×B – C
Hint #7
Substitute 2×B – C for A and F (from eq.4a), 3×B – C for D (from eq.2c), and C for E into eq.1: 2×B – C + B + C + 3×B – C + C + 2×B – C = 33 which simplifies to 8×B – C = 33 In the above equation, add C to both sides, and subtract 33 from each side: 8×B – C + C – 33 = 33 + C – 33 which makes eq.1a) 8×B – 33 = C
Hint #8
To make eq.1a true, check several possible values for B and C: If B = 5 then C = 8×5 – 33 = 40 – 33 = 7 If B = 6 then C = 8×6 – 33 = 48 – 33 = 15 If B > 6 then C > 15 If B = 4 then C = 8×4 – 33 = 32 – 33 = –1 If B < 4 then C < –1 Since B and C must be one-digit non-negative integers, then B = 5 and C = 7 which also makes E = C = 7
Hint #9
Substitute 5 for B, and 7 for C in eq.4a: A = F = 2×5 – 7 which becomes A = F = 10 – 7 which makes A = F = 3
Solution
Substitute 5 for B, and 7 for C in eq.2b: D = 3×5 – 7 which becomes D = 15 – 7 which makes D = 8 and makes ABCDEF = 357873