Puzzle for March 16, 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 and EF are 2-digit numbers (not D×E or E×F).
Scratchpad
Help Area
Hint #1
In eq.5, replace A with D - F (from eq.2): C + D = D - F + B + F which becomes C + D = D + B Subtract D from both sides of the above equation: C + D - D = D + B - D which makes C = B
Hint #2
In eq.3, replace B with C: E + F - C = A + C + C which becomes E + F - C = A + 2×C Add C to both sides of the equation above: E + F - C + C = A + 2×C + C which becomes eq.3a) E + F = A + 3×C
Hint #3
Subtract F from both sides of eq.4: D + F - F = B + E - F which becomes D = B + E - F In eq.5, substitute B + E - F for D: C + B + E - F = A + B + F Add (F - B) to both sides of the above equation: C + B + E - F + (F - B) = A + B + F + (F - B) which becomes eq.5a) C + E = A + 2×F
Hint #4
Subtract the left and right sides of eq.5a from the left and right sides of eq.3a, respectively: E + F - (C + E) = A + 3×C - (A + 2×F) which is equivalent to E + F - C - E = A + 3×C - A - 2×F which becomes F - C = 3×C - 2×F Add C + 2×F to each side of the above equation: F - C + C + 2×F = 3×C - 2×F + C + 2×F which simplifies to 3×F = 4×C Divide both sides by 4: 3×F ÷ 4 = 4×C ÷ 4 which makes ¾×F = C which also makes B = C = ¾×F
Hint #5
In eq.4, substitute ¾×F for B: D + F = ¾×F + E Subtract ¾×F from each side of the equation above: D + F - ¾×F = ¾×F + E - ¾×F which becomes eq.4a) D + ¼×F = E
Hint #6
eq.6 may be re-written as: B + C + 10×D + E = 10×E + F Substitute ¾×F for both B and C in the above equation: ¾×F + ¾×F + 10×D + E = 10×E + F which becomes 1½×F + 10×D + E = 10×E + F Subtract both E and F from each side: 1½×F + 10×D + E - E - F = 10×E + F - E - F which becomes eq.6a) ½×F + 10×D = 9×E
Hint #7
Substitute (D + ¼×F) for E (from eq.4a) in eq.6a: ½×F + 10×D = 9×(D + ¼×F) which is equivalent to ½×F + 10×D = 9×D + 2¼×F Subtract ½×F and 9×D from both sides: ½×F + 10×D - ½×F - 9×D = 9×D + 2¼×F - ½×F - 9×D which simplifies to D = 1¾×F
Hint #8
Substitute 1¾×F for D in eq.4a: 1¾×F + ¼×F = E which makes 2×F = E
Hint #9
Substitute 1¾×F for D in eq.2: A = 1¾×F - F which makes A = ¾×F
Solution
Substitute ¾×F for A and B and C, 1¾×F for D, and 2×F for E in eq.1: ¾×F + ¾×F + ¾×F + 1¾×F + 2×F + F = 28 which simplifies to 7×F = 28 Divide both sides by 7: 7×F ÷ 7 = 28 ÷ 7 which means F = 4 making A = B = C = ¾×F = ¾ × 4 = 3 D = 1¾×F = 1¾ × 4 = 7 E = 2×F = 2 × 4 = 8 and ABCDEF = 333784