Puzzle for June 23, 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.
* DE is a 2-digit number (not D×E).
Scratchpad
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Hint #1
In eq.2, replace C with B + D (from eq.4): E + F = B + D + D which becomes E + F = B + 2×D In the above equation, replace B with A + F (from eq.3): E + F = A + F + 2×D Subtract F from each side: E + F - F = A + F + 2×D - F which becomes eq.2a) E = A + 2×D
Hint #2
eq.5 may be written as: 10×D + E - A = A + C In the above equation, replace E with A + 2×D (from eq.2a): 10×D + A + 2×D - A = A + C which becomes eq.5a) 12×D = A + C
Hint #3
In eq.4, substitute A + F for B (from eq.3): eq.4a) C = A + F + D Add A and D to both sides of eq.2: E + F + A + D = C + D + A + D which may be written as E + A + F + D = A + C + D + D Substitute C for A + F + D (from eq.4a), and 12×D for A + C (from eq.5a) in the above equation: E + C = 12×D + D + D which becomes eq.4b) E + C = 14×D
Hint #4
Substitute B + D for C (from eq.4) in eq.4b: E + B + D = 14×D Subtract D from both sides: E + B + D - D = 14×D - D which becomes E + B = 13×D which may be written as eq.2b) B + E = 13×D
Hint #5
eq.1 may be written as: A + C + B + E + D + F = 31 Substitute 12×D for A + C (from eq.5a), and 13×D for B + E (from eq.2b) in the equation above: 12×D + 13×D + D + F = 31 which becomes 26×D + F = 31 Subtract 26×D from both sides: 26×D + F - 26×D = 31 - 26×D which means eq.1a) F = 31 - 26×D
Hint #6
To make eq.1a true, check several possible values for D and F: If D = 0, then F = 31 - 26×0 = 31 - 0 = 31 If D = 1, then F = 31 - 26×1 = 31 - 26 = 5 If D = 2, then F = 31 - 26×2 = 31 - 52 = -21 If D > 2, then F < -21 Since F must be a one-digit non-negative integer, then F = 5 making D = 1
Hint #7
Substitute 1 for D in eq.2b: B + E = 13×1 which means B + E = 13 Subtract B from each side: B + E - B = 13 - B which becomes eq.2c) E = 13 - B
Hint #8
Substitute 1 for D in eq.4: eq.4c) C = B + 1 Substitute 13 - B for E (from eq.2c), 5 for F, B + 1 for C (from eq.4c), and 1 for D in eq.2: 13 - B + 5 = B + 1 + 1 which becomes 18 - B = B + 2 Add B to each side, and subtract 2 from each side: 18 - B + B - 2 = B + 2 + B - 2 which means 16 = 2×B Divide both sides by 2: 16 ÷ 2 = 2×B ÷ 2 which means 8 = B
Hint #9
Substitute 8 for B in eq.4c: C = 8 + 1 which makes C = 9
Hint #10
Substitute 8 for B in eq.2c: 13 - 8 = E which means 5 = E
Solution
Substitute 8 for B, and 5 for F in eq.3: 8 = A + 5 Subtract 5 from each side: 8 - 5 = A + 5 - 5 which means 3 = A making ABCDEF = 389155