Puzzle for March 23, 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.
Scratchpad
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Hint #1
In eq.6, replace B + C with E + F (from eq.2): E + F + D - F = A + E which becomes E + D = A + E Subtract E from both sides of the above equation: E + D - E = A + E - E which makes D = A
Hint #2
Subtract both B and F from each side of eq.2: E + F - B - F = B + C - B - F which becomes eq.2a) E - B = C - F Subtract both C and F from each side of eq.4: A + F - C - F = C + E - C - F which becomes eq.4a) A - C = E - F
Hint #3
In eq.3, replace D with A: A - C = B + F In the equation above, replace A - C with E - F (from eq.4a): E - F = B + F Add (F - B) to each side: E - F + (F - B) = B + F + (F - B) which becomes E - B = 2×F Substitute C - F for E - B (from eq.2a): C - F = 2×F Add F to both sides: C - F + F = 2×F + F which means 3×F = C
Hint #4
Subtract both C and E from each side of eq.5: C + D - C - E = B + E - C - E which becomes eq.5a) D - E = B - C
Hint #5
Substitute D for A in eq.4: D + F = C + E Subtract both E and F from each side of the above equation: D + F - E - F = C + E - E - F which becomes eq.4b) D - E = C - F
Hint #6
Substitute B - C for D - E (from eq.5a) in eq.4b: B - C = C - F Add C to both sides: B - C + C = C - F + C which becomes B = 2×C - F Substitute (3×F) for C: B = 2×(3×F) - F which becomes B = 6×F - F which means B = 5×F
Hint #7
Substitute 5×F for B, and 3×F for C in eq.2: E + F = 5×F + 3×F Subtract F from each side of the above equation: E + F - F = 5×F + 3×F - F which means E = 7×F
Hint #8
Substitute 5×F for B, and 3×F for C in eq.3: D - 3×F = 5×F + F Add 3×F to each side of the above equation: D - 3×F + 3×F = 5×F + F + 3×F which means D = 9×F which also makes A = D = 9×F
Solution
Substitute 9×F for A and D, 5×F for B, 3×F for C, and 7×F for E in eq.1: 9×F + 5×F + 3×F + 9×F + 7×F + F = 34 which simplifies to 34×F = 34 Divide both sides by 34: 34×F ÷ 34 = 34 ÷ 34 which makes F = 1 making A = D = 9×F = 9 × 1 = 9 B = 5×F = 5 × 1 = 5 C = 3×F = 3 × 1 = 3 E = 7×F = 7 × 1 = 7 and ABCDEF = 953971