Puzzle for December 22, 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.
* BC and EF are 2-digit numbers (not B×C or E×F).
Scratchpad
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Hint #1
Add C + E to both sides of eq.2: F - E + C + E = D - C + C + E which becomes F + C = D + E In eq.5, replace D + E with F + C: A + B + C = F + C Subtract C from each side: A + B + C - C = F + C - C which becomes eq.5a) A + B = F
Hint #2
Add A to both sides of eq.3: B - A + A = E + A which becomes eq.3a) B = A + E Add C to both sides of eq.4: B - C + C = D + C which becomes B = D + C which is the same as eq.4a) B = C + D
Hint #3
eq.6 may be written as: A + 10×B + C = D + 10×E + F Substitute (A + E) for B (from eq.3a) in the above equation: A + 10×(A + E) + C = D + 10×E + F which is equivalent to A + 10×A + 10×E + C = D + 10×E + F Subtract 10×E from each side: A + 10×A + 10×E + C - 10×E = D + 10×E + F - 10×E which becomes eq.6a) 11×A + C = D + F
Hint #4
In eq.6a, substitute A + B for F (from eq.5a): 11×A + C = D + A + B Subtract A from each side: 11×A + C - A = D + A + B - A which becomes eq.6b) 10×A + C = D + B
Hint #5
In eq.6b, replace B with C + D (from eq.4a): 10×A + C = D + C + D Subtract C from each side: 10×A + C - C = D + C + D - C which simplifies to 10×A = 2×D Divide both sides by 2: 10×A ÷ 2 = 2×D ÷ 2 which makes 5×A = D
Hint #6
Substitute A + E for B (from eq.3a), and 5×A for D in eq.5: A + A + E + C = 5×A + E Subtract (2×A + E) from each side: A + A + E + C - (2×A + E) = 5×A + E - (2×A + E) which is the same as A + A + E + C - 2×A - E = 5×A + E - 2×A - E which simplifies to C = 3×A
Hint #7
Substitute 3×A for C, and 5×A for D in eq.4a: B = 3×A + 5×A = 8×A
Hint #8
In eq.5a, replace B with 8×A: A + 8×A = F which means 9×A = F
Hint #9
Substitute 8×A for B in eq.3: 8×A - A = E which becomes 7×A = E
Solution
Substitute 8×A for B, 3×A for C, 5×A for D, 7×A for E, and 9×A for F in eq.1: A + 8×A + 3×A + 5×A + 7×A + 9×A = 33 which simplifies to 33×A = 33 Divide both sides by 33: 33×A ÷ 33 = 33 ÷ 33 which makes A = 1 making B = 8×A = 8 × 1 = 8 C = 3×A = 3 × 1 = 3 D = 5×A = 5 × 1 = 5 E = 7×A = 7 × 1 = 7 F = 9×A = 9 × 1 = 9 and ABCDEF = 183579