Puzzle for January 16, 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.
* AB and CD are 2-digit numbers (not A×B or C×D).
Scratchpad
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Hint #1
In eq.2, replace C with D + F (from eq.4): A + D + F = E + F Subtract F from both sides of the above equation: A + D + F - F = E + F - F which becomes A + D = E In eq.3, replace E with A + D: A + D = B + D Subtract D from each side: A + D - D = B + D - D which becomes A = B
Hint #2
Subtract D from each side of eq.4: C - D = D + F - D which becomes eq.4a) C - D = F Subtract F from each side of eq.4: C - F = D + F - F which becomes eq.4b) C - F = D
Hint #3
In eq.5, replace C - D with F (from eq.4a): A - F = F Add F to both sides of the above equation: A - F + F = F + F which makes A = 2×F
Hint #4
eq.6 may be expressed as: 10×A + B - F = 10×C + D + E + F In the above equation, substitute B + D for E (from eq.3): 10×A + B - F = 10×C + D + B + D + F Substitute (2×F) for both A and B, and C - F for D (from eq.4b): 10×(2×F) + 2×F - F = 10×C + C - F + 2×F + C - F + F which simplifies to 20×F + F = 12×C + F Subtract F from each side: 20×F + F - F = 12×C + F - F which makes 20×F = 12×C Divide both sides by 12: 20×F ÷ 12 = 12×C ÷ 12 which means 1⅔×F = C
Hint #5
Substitute 1⅔×F for C in eq.4: 1⅔×F = D + F Subtract F from each side: 1⅔×F - F = D + F - F which makes ⅔×F = D
Hint #6
Substitute 2×F for B, and ⅔×F for D in eq.3: E = 2×F + ⅔×F which makes E = 2⅔×F
Solution
Substitute 2×F for A and B, 1⅔×F for C, ⅔×F for D, and 2⅔×F for E in eq.1: 2×F + 2×F + 1⅔×F + ⅔×F + 2⅔×F + F = 30 which simplifies to 10×F = 30 Divide both sides by 10: 10×F ÷ 10 = 30 ÷ 10 which means F = 3 making A = B = 2×F = 2 × 3 = 6 C = 1⅔×F = 1⅔ × 3 = 5 D = ⅔×F = ⅔ × 3 = 2 E = 2⅔×F = 2⅔ × 3 = 8 and ABCDEF = 665283