Puzzle for September 9, 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
In eq.2, replace E + F with C (from eq.5): D + C = A + C Subtract C from both sides of the above equation: D + C – C = A + C – C which makes D = A
Hint #2
In eq.4, replace A with D: D + E = D + F Subtract D from each side of the equation above: D + E – D = D + F – D which makes E = F
Hint #3
eq.2 may be written as: D + F + E = A + C In the above equation, substitute A + E for D + F (from eq.4): A + E + E = A + C which becomes A + 2×E = A + C Subtract A from each side: A + 2×E – A = A + C – A which makes 2×E = C
Hint #4
eq.1 may be re-written as: A + D + B + C + E + F = 20 Substitute A + D for B + C + E + F (from eq.6) in the above equation: A + D + A + D = 20 Substitute A for D: A + A + A + A = 20 which means 4×A = 20 Divide both sides by 4: 4×A ÷ 4 = 20 ÷ 4 which makes A = 5 and which also means D = A = 5
Hint #5
Substitute 2×E for C, and 5 for D in eq.3: B – 2×E = 5 – E Add 2×E to each side: B – 2×E + 2×E = 5 – E + 2×E which becomes eq.3a) B = 5 + E
Solution
Substitute 5 for A and D, 5 + E for B (from eq.3a), 2×E for C, and E for F in eq.1: 5 + 5 + E + 2×E + 5 + E + E = 20 which simplifies to 15 + 5×E = 20 Subtract 15 from both sides of the above equation: 15 + 5×E – 15 = 20 – 15 which makes 5×E = 5 Divide both sides by 5: 5×E ÷ 5 = 5 ÷ 5 which means E = 1 making B = 5 + E = 5 + 1 = 6 (from eq.3a) C = 2×E = 2 × 1 = 2 F = E = 1 and ABCDEF = 562511