Puzzle for June 20, 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.
* CD, DE, and EF are 2-digit numbers (not C×D, D×E, or E×F).
Scratchpad
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Hint #1
In eq.1, replace A + B + C + D + E with F - E (from eq.3): F - E + F = 17 which becomes 2×F - E = 17 In the above equation, add E to both sides, and subtract 17 from both sides: 2×F - E + E - 17 = 17 + E - 17 which becomes 2×F - 17 = E which may be written as eq.1a) E = 2×F - 17
Hint #2
To make eq.1a true, check several possible values for F and E: If F = 9, then E = 2×9 - 17 = 18 - 17 = 1 If F = 8, then E = 2×8 - 17 = 16 - 17 = -1 If F < 8, then E < -1 Since F must be a non-negative integer, then F = 9 which makes E = 1
Hint #3
In eq.3, replace A + B + C with D + E (from eq.2): D + E + D + E = F - E which becomes 2×D + 2×E = F - E In the above equation, replace E with 1, and F with 9: 2×D + 2×1 = 9 - 1 which becomes 2×D + 2 = 8 Subtract 2 from each side: 2×D + 2 - 2 = 8 - 2 which makes 2×D = 6 Divide both sides by 2: 2×D ÷ 2 = 6 ÷ 2 which means D = 3
Hint #4
Substitute 3 for D, and 1 for E in eq.2: 3 + 1 = A + B + C Subtract C from both sides: 3 + 1 - C = A + B + C - C which becomes eq.2a) 4 - C = A + B
Hint #5
Substitute 9 for F, 3 for D, and 4 - C for A + B (from eq.2a) in eq.4: 9 - 3 = 4 - C + 3 which becomes 6 = 7 - C Subtract 6 from both sides, and add C to both sides: 6 - 6 + C = 7 - C - 6 + C which makes C = 1
Hint #6
eq.5 may be written as: 10×C + D + 10×E + F = B + 10×D + E Subtract D and E from both sides of the above equation: 10×C + D + 10×E + F - D - E = B + 10×D + E - D - E which simplifies to 10×C + 9×E + F = B + 9×D Substitute 1 for C and E, 9 for F, and 3 for D: 10×1 + 9×1 + 9 = B + 9×3 which becomes 10 + 9 + 9 = B + 27 which makes 28 = B + 27 Subtract 27 from both sides: 28 - 27 = B + 27 - 27 which means 1 = B
Solution
Substitute 1 for B and C in eq.2a: 4 - 1 = A + 1 which means 3 = A + 1 Subtract 1 from both sides: 3 - 1 = A + 1 - 1 which makes 2 = A and ABCDEF = 211319