Puzzle for November 17, 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
eq.1 may be written as: A + D + E + F + B + C = 25 In the above equation, replace A + D + E + F with B (from eq.2): B + B + C = 25 which becomes 2×B + C = 25 Subtract 2×B from each side: 2×B + C – 2×B = 25 – 2×B which becomes eq.1a) C = 25 – 2×B
Hint #2
To make eq.1a true, check several possible values for B and C: If B = 9, then C = 25 – 2×9 = 25 – 18 = 7 If B = 8, then C = 25 – 2×8 = 25 – 16 = 9 If B = 7, then C = 25 – 2×7 = 25 – 14 = 11 If B < 7, then C > 11 Since C must be a one-digit integer, then: B = 8 and C = 9 or: B = 9 and C = 7
Hint #3
Begin checking: B = 8, and C = 9 ... Substituting 9 for C in eq.3 would yield: A = 9 – E Adding E to both sides of the above equation would make: A + E = 9 – E + E which would make eq.3a) A + E = 9
Hint #4
Continue checking: B = 8, and C = 9 ... eq.2 could be re-written as: B = A + E + D + F Substituting 8 for B, and 9 for A + E (from eq.3a) in the equation above would yield: 8 = 9 + D + F Subtracting 9 from both sides would yield: 8 – 9 = 9 + D + F – 9 which would make eq.2a) –1 = D + F
Hint #5
Finish checking: B = 8, and C = 9 ... Since D and F are non-negative integers, then D + F ≥ 0 which means –1 ≠ D + F (from eq.2a) which means that B ≠ 8 and C ≠ 9 and makes B = 9 and C = 7
Hint #6
Substitute 7 for C, and 9 for B in eq.5: 7 + D + E = 9 + F Subtract 7 from both sides of the equation above: 7 + D + E – 7 = 9 + F – 7 which becomes eq.5a) D + E = 2 + F
Hint #7
Substitute 9 for B, and 2 + F for D + E (from eq.5a) in eq.2: 9 = A + 2 + F + F which becomes 9 = A + 2 + 2×F Subtract 2 and 2×F from both sides of the equation above: 9 – 2 – 2×F = A + 2 + 2×F – 2 – 2×F which becomes eq.2b) 7 – 2×F = A
Hint #8
Substitute 7 – 2×F for A (from eq.2b), and 7 for C in eq.3: 7 – 2×F = 7 – E Subtract 7 from each side of the above equation: 7 – 2×F – 7 = 7 – E – 7 which becomes –2×F = –E which means 2×F = E
Hint #9
Substitute 2×F for E in eq.5a: D + 2×F = 2 + F Subtract 2×F from both sides of the equation above: D + 2×F – 2×F = 2 + F – 2×F eq.5b) D = 2 – F
Solution
Substitute 2×F for E, 7 – 2×F for A (from eq.2b), and 7 for C in eq.4: 2×F + F = 7 – 2×F + 7 – 2×F which becomes 3×F = 14 – 4×F Add 4×F to both sides of the above equation: 3×F + 4×F = 14 – 4×F + 4×F which makes 7×F = 14 Divide both sides by 7: 7×F ÷ 7 = 14 ÷ 7 which means F = 2 making A = 7 – 2×F = 7 – 2×2 = 7 – 4 = 3 (from eq.2b) D = 2 – F = 2 – 2 = 0 (from eq.5b) E = 2×F = 2×2 = 4 and ABCDEF = 397042