Puzzle for December 29, 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.
* AB, DE, and EF are 2-digit numbers (not A×B, D×E, or E×F).
Scratchpad
Help Area
Hint #1
In eq.5, substitute D + E + F for A + B (from eq.2): D + E + F + F = C + D + E Subtract both D and E from the equation above: D + E + F + F - D - E = C + D + E - D - E which makes 2×F = C
Hint #2
eq.4 may be written as: B + C = A + D + E Subtract the left and right sides of the above equation from the respective sides of eq.5: A + B + F - (B + C) = C + D + E - (A + D + E) which is equivalent to A + B + F - B - C = C + D + E - A - D - E which becomes A + F - C = C - A Add A + C to both sides of the above equation: A + F - C + A + C = C - A + A + C which becomes eq.5a) 2×A + F = 2×C
Hint #3
In eq.5a, substitute (2×F) for C: 2×A + F = 2×(2×F) which becomes 2×A + F = 4×F Subtract F from both sides: 2×A + F - F = 4×F - F which means 2×A = 3×F Divide both sides by 2: 2×A ÷ 2 = 3×F ÷ 2 which means A = 1½×F
Hint #4
eq.6 may be written as: D + 10×E + F = 10×A + B + C + 10×D + E Subtract both D and E from each side of the above equation: D + 10×E + F - D - E = 10×A + B + C + 10×D + E - D - E which becomes eq.6a) 9×E + F = 10×A + B + C + 9×D
Hint #5
In eq.6a, replace E with C + D (from eq.3): 9×(C + D) + F = 10×A + B + C + 9×D which is equivalent to 9×C + 9×D + F = 10×A + B + C + 9×D Subtract both C and 9×D from each side: 9×C + 9×D + F - C - 9×D = 10×A + B + 9×D + C - C - 9×D which becomes 8×C + F = 10×A + B Substitute (1½×F) for A, and (2×F) for C: 8×(2×F) + F = 10×(1½×F) + B which becomes 16×F + F = 15×F + B Subtract 15×F from both sides: 16×F + F - 15×F = 15×F + B - 15×F which means 2×F = B
Hint #6
In eq.4, replace E with C + D (from eq.3): A + D + C + D = B + C which becomes A + 2×D + C = B + C Subtract C from both sides of the above equation: A + 2×D + C - C = B + C - C which becomes A + 2×D = B Substitute 1½×F for A, and 2×F for B: 1½×F + 2×D = 2×F Subtract 1½×F from both sides: 1½×F + 2×D - 1½×F = 2×F - 1½×F which simplifies to 2×D = ½×F Divide both sides by 2: 2×D ÷ 2 = ½×F ÷ 2 which means D = ¼×F
Hint #7
Substitute 2×F for C, and ¼×F for D in eq.3: 2×F + ¼×F = E which means 2¼×F = E
Solution
Substitute 1½×F for A, 2×F for B and C, ¼×F for D, and 2¼×F for E in eq.1: 1½×F + 2×F + 2×F + ¼×F + 2¼×F + F = 36 which simplifies to 9×F = 36 Divide both sides by 9: 9×F ÷ 9 = 36 ÷ 9 which means F = 4 making A = 1½×F = 1½ × 4 = 6 B = C = 2×F = 2 × 4 = 8 D = ¼×F = ¼ × 4 = 1 E = 2¼×F = 2¼ × 4 = 9 and ABCDEF = 688194