Puzzle for October 12, 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
Add D to both sides of eq.4: C – D + D = A + F + D which becomes C = A + F + D In eq.5, substitute A + F + D for C: B + E = A + F + D which may be written as eq.5a) A + F + D = B + E
Hint #2
Subtract the left and right sides of eq.5a from the left and right sides of eq.2, respectively: D + E + F – (A + F + D) = A + B – (B + E) which is the same as D + E + F – A – F – D = A + B – B – E which simplifies to E – A = A – E Add both A and E to each side of the above equation: E – A + A + E = A – E + A + E which simplifies to 2×E = 2×A Divide both sides by 2: 2×E ÷ 2 = 2×A ÷ 2 which makes E = A
Hint #3
In eq.6, replace A with E: E ÷ E = B – E which becomes 1 = B – E Add E to both sides of the above equation: 1 + E = B – E + E which makes eq.6a) 1 + E = B
Hint #4
In eq.5, replace B with 1 + E (from eq.6a): 1 + E + E = C which makes eq.5a) 1 + 2×E = C
Hint #5
In eq.3, substitute 1 + 2×E for C (from eq.5a): E + F = 1 + 2×E + D Subtract E from both sides of the equation above: E + F – E = 1 + 2×E + D – E which becomes eq.3a) F = 1 + E + D
Hint #6
In eq.4, substitute 1 + 2×E for C (from eq.5a), and E for A: 1 + 2×E – D = E + F Subtract E from both sides: 1 + 2×E – D – E = E + F – E which becomes eq.4a) 1 + E – D = F
Hint #7
Substitute 1 + E – D for F (from eq.4a) in eq.3a: 1 + E – D = 1 + E + D Subtract 1 and E from both sides of the above equation: 1 + E – D – 1 – E = 1 + E + D – 1 – E which simplifies to –D = D which means 0 = D
Hint #8
Substitute 0 for D in eq.3a: F = 1 + E + 0 which is the same as eq.3b) F = 1 + E
Solution
Substitute E for A, 1 + E for B (from eq.6a), 1 + 2×E for C (from eq.5a), 0 for D, and 1 + E for F (from eq.3b) in eq.1: E + 1 + E + 1 + 2×E + 0 + E + 1 + E = 27 which simplifies to 6×E + 3 = 27 Subtract 3 from both sides of the above equation: 6×E + 3 – 3 = 27 – 3 which makes 6×E = 24 Divide both sides by 6: 6×E ÷ 6 = 24 ÷ 6 which means E = 4 making A = E = 4 B = 1 + E = 1 + 4 = 5 (from eq.6a) C = 1 + 2×E = 1 + 2×4 = 1 + 8 = 9 (from eq.5a) F = 1 + E = 1 + 4 = 5 (from eq.3b) and ABCDEF = 459045