Puzzle for March 18, 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, add A to both sides, and subtract E from both sides: E – F + A – E = D – A + A – E which becomes –F + A = D – E In eq.4, replace D – E with –F + A: –F + A = A – B Subtract A from both sides of the equation above: –F + A – A = A – B – A which makes –F = –B which means F = B
Hint #2
In eq.5, replace F with B: A + D – B = B + B which becomes A + D – B = 2B Add B to each side of the above equation: A + D – B + B = 2×B + B which becomes eq.5a) A + D = 3×B
Hint #3
In eq.6, replace F with B: A – B = B + D – A In the above equation, add A to both sides, and subtract B from each side: A – B + A – B = B + D – A + A – B which becomes eq.6a) 2×A – 2×B = D
Hint #4
In eq.5a, substitute 2×A – 2×B for D (from eq.6a): A + 2×A – 2×B = 3×B which becomes 3×A – 2×B = 3×B Add 2×B to each side of the equation above: 3×A – 2×B + 2×B = 3×B + 2×B which makes 3×A = 5×B Divide both sides by 3: 3×A ÷ 3 = 5×B ÷ 3 which makes A = 1⅔×B
Hint #5
Substitute (1⅔×B) for A in eq.6a: 2×(1⅔×B) – 2×B = D which becomes 3⅓×B – 2×B = D which makes 1⅓×B = D
Hint #6
Substitute B for F, 1⅓×B for D, and 1⅔×B for A in eq.2: E – B = 1⅓×B – 1⅔×B which becomes E – B = –⅓×B Add B to both sides of the above equation: E – B + B = –⅓×B + B which makes E = ⅔×B
Hint #7
Substitute B for F, 1⅓×B for D, and ⅔×B for E in eq.3: B + C + B = 1⅓×B + ⅔×B which becomes 2×B + C = 2×B Subtract 2×B from both sides of the above equation: 2×B + C – 2×B = 2×B – 2×B which makes C = 0
Solution
Substitute 1⅔×B for A, 0 for C, 1⅓×B for D, ⅔×B for E, and B for F in eq.1: 1⅔×B + B + 0 + 1⅓×B + ⅔×B + B = 17 which simplifies to 5⅔×B = 17 Divide both sides of the above equation by 5⅔: 5⅔×B ÷ 5⅔ = 17 ÷ 5⅔ which means B = 3 making A = 1⅔×B = 1⅔ × 3 = 5 D = 1⅓×B = 1⅓ × 3 = 4 E = ⅔×B = ⅔ × 3 = 2 F = B = 3 and ABCDEF = 530423