Puzzle for October 5, 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
Subtract C from both sides of eq.3: B – C = C + E – C which becomes eq.3a) B – C = E In eq.2, substitute B + C for D (from eq.5), and (B – C) for E (from eq.3a): A = B + C – (B – C) which is equivalent to A = B + C – B + C which becomes eq.3b) A = 2×C
Hint #2
In eq.4, substitute (B + C) for D (from eq.5): C = F – (B + C) which is equivalent to C = F – B – C Add B and C to each side of the equation above: C + B + C = F – B – C + B + C which becomes eq.4a) B + 2×C = F
Hint #3
Substitute 2×C for A, B + C for D (from eq.5), B – C for E (from eq.3a), and B + 2×C for F (from eq.4a) in eq.1: 2×C + B + C + B + C + B – C + B + 2×C = 21 which becomes 4×B + 5×C = 21 Subtract 5×C from both sides of the above equation: 4×B + 5×C – 5×C = 21 – 5×C which becomes 4×B = 21 – 5×C Divide both sides by 4: 4×B ÷ 4 = (21 – 5×C) ÷ 4 which becomes eq.1a) B = (21 – 5×C) ÷ 4
Hint #4
To make eq.1a true, check several possible values for C and B: If C = 0, then B = (21 – 5×0) ÷ 4 = (21 – 0) ÷ 4 = 21 ÷ 4 = 5¼ If C = 1, then B = (21 – 5×1) ÷ 4 = (21 – 5) ÷ 4 = 16 ÷ 4 = 4 If C = 2, then B = (21 – 5×2) ÷ 4 = (21 – 10) ÷ 4 = 11 ÷ 4 = 2¾ If C = 3, then B = (21 – 5×3) ÷ 4 = (21 – 15) ÷ 4 = 6 ÷ 4 = 1½ If C = 4, then B = (21 – 5×4) ÷ 4 = (21 – 20) ÷ 4 = 1 ÷ 4 = ¼ If C = 5, then B = (21 – 5×5) ÷ 4 = (21 – 25) ÷ 4 = –4 ÷ 4 = –1 If C > 5, then B < –1 Since B must be a one-digit non-negative integer, then B = 4 which means C = 1
Hint #5
Substitute 1 for C in eq.3b: A = 2×1 which means A = 2
Hint #6
Substitute 4 for B, and 1 for C in eq.4a: 4 + 2×1 = F which becomes 4 + 2 = F which makes 6 = F
Hint #7
Substitute 4 for B, and 1 for C in eq.5: D = 4 + 1 which means D = 5
Solution
Substitute 4 for B, and 1 for C in eq.3a: 4 – 1 = E which makes 3 = E and ABCDEF = 241536