Puzzle for December 21, 2018 ( )
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 and EF are 2-digit numbers (not C×D or E×F).
Scratchpad
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Hint #1
eq.6 may be written as: B + 10×C + D = 10×E + F which is the same as eq.6a) B + D + 10×C = 10×E + F Add C + D to both sides of eq.3: B - C + C + D = F - E - D + C + D which becomes eq.3a) B + D = F - E + C
Hint #2
In eq.6a, substitute F - E + C for B + D (from eq.3a): F - E + C + 10×C = 10×E + F Add (E - F) to both sides of the above equation: F - E + C + 10×C + (E - F) = 10×E + F + (E - F) which simplifies to 11×C = 11×E Divide both sides by 11: 11×C ÷ 11 = 11×E ÷ 11 which means C = E
Hint #3
In eq.2, replace E with C: A + B + C = C + F Subtract C from both sides of the equation above: A + B + C - C = C + F - C which becomes eq.2a) A + B = F
Hint #4
In eq.3, replace E with C: B - C = F - C - D Add C + D to both sides of the equation above: B - C + C + D = F - C - D + C + D which becomes B + D = F Substitute A + B for F (from eq.2a) into the equation above: B + D = A + B Subtract B from both sides: B + D - B = A + B - B which makes D = A
Hint #5
Replace D with A in eq.4: A + A = C which makes 2×A = C
Hint #6
Substitute A for D, and 2×A for both C and E in eq.5: B - 2×A = 2×A + A + 2×A = 5×A Add 2×A to both sides of the equation above: B - 2×A + 2×A = 5×A + 2×A which means B = 7×A
Hint #7
Substitute 7×A for B in eq.2a: A + 7×A = F which makes 8×A = F
Solution
Substitute 7×A for B, 2×A for C and E, A for D, and 8×A for F in eq.1: A + 7×A + 2×A + A + 2×A + 8×A = 21 which simplifies to 21×A = 21 Divide both sides of the above equation by 21: 21×A ÷ 21 = 21 ÷ 21 which makes A = 1 making B = 7×A = 7 × 1 = 7 C = E = 2×A = 2 × 1 = 2 D = A = 1 F = 8×A = 8 × 1 = 8 and ABCDEF = 172128