Puzzle for November 22, 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 E to both sides of eq.2: C – D – E + E = A + F + E which becomes eq.2a) C – D = A + E + F In eq.2a, replace E + F with B (from eq.4): eq.2b) C – D = A + B
Hint #2
In eq.2b, replace A + B with C – B (from eq.3): C – D = C – B Subtract C from each side of the equation above: C – D – C = C – B – C which makes –D = –B which means D = B
Hint #3
In eq.6, substitute B for D: E = B ÷ B which means E = 1
Hint #4
Add D to both sides of eq.2a: C – D + D = A + F + E + D which becomes eq.2c) C = A + F + E + D eq.1 may be written as: eq.1a) A + F + E + D + B + C = 17
Hint #5
Substitute C for A + F + E + D (from eq.2c) in eq.1a: C + B + C = 17 which becomes eq.1b) B + 2×C = 17 Subtract 2×C from both sides of eq.1b: B + 2×C – 2×C = 17 – 2×C which makes B = 17 – 2×C and also makes eq.1c) D = B = 17 – 2×C
Hint #6
Substitute (17 – 2×C) for B (from eq.1c) in eq.3: A + (17 – 2×C) = C – (17 – 2×C) which is equivalent to A + 17 – 2×C = C – 17 + 2×C which becomes A + 17 – 2×C = 3×C – 17 In the above equation, subtract 17 from both sides, and add 2×C to both sides: A + 17 – 2×C – 17 + 2×C = 3×C – 17 – 17 + 2×C which makes eq.3a) A = 5×C – 34
Hint #7
Substitute D for B in eq.4: D = E + F Subtract E from each side of the equation above: D – E = E + F – E which becomes eq.4a) D – E = F
Hint #8
Substitute F for D – E (from eq.4a) in eq.5: eq.5a) F = C × F Since C and F are non-negative integers, eq.5a makes either C = 1 or F = 0
Hint #9
Check: C = 1 ... Substituting 1 for C in eq.3a would yield: A = 5×1 – 34 which would make A = 5 – 34 which would make A = –29 Since A must be a non-negative one-digit integer, then A ≠ –29 which means C ≠ 1 and therefore makes F = 0
Hint #10
Substitute 1 for E, and 0 for F in eq.4: B = 1 + 0 which makes B = 1 and also makes D = B = 1
Hint #11
Substitute 1 for B in eq.1b: 1 + 2×C = 17 Subtract 1 from both sides of the equation above: 1 + 2×C – 1 = 17 – 1 which makes 2×C = 16 Divide both sides by 2: 2×C ÷ 2 = 16 ÷ 2 which makes C = 8
Solution
Substitute 8 for C in eq.3a: A = 5×8 – 34 which becomes A = 40 – 34 which means A = 6 making ABCDEF = 618110