Puzzle for April 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.
Scratchpad
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Hint #1
In eq.5, replace C with A - F (from eq.6): B - F = A - F + E Add (F - E) to each side of the equation above: B - F + (F - E) = A - F + E + (F - E) which simplifies to eq.5a) B - E = A
Hint #2
In eq.2, replace A with B - E (from eq.5a): B - E = D - E Add E to each side of the above equation: B - E + E = D - E + E which means B = D
Hint #3
Add F to each side of eq.5: B - F + F = C + E + F which becomes B = C + E + F Substitute D for B, and C + D for E + F (from eq.4) in the equation above: D = C + C + D which becomes D = 2×C + D Subtract D from each side: D - D = 2×C + D - D which makes 0 = 2×C which means 0 = C
Hint #4
In eq.6, replace C with 0, and add F to each side: 0 + F = A - F + F which makes F = A
Hint #5
Substitute (B - E) for both A (from eq.2) and for F in eq.3: E - (B - E) = (B - E) - B which may be written as E - B + E = B - E - B which becomes 2×E - B = -E Add E + B to each side: 2×E - B + E + B = -E + E + B which makes 3×E = B and which also makes D = B = 3×E
Hint #6
Substitute 3×E for D in eq.2: A = 3×E - E which makes A = 2×E and which also makes F = A = 2×E
Solution
Substitute 2×E for A and F, 3×E for B and D, and 0 for C in eq.1: 2×E + 3×E + 0 + 3×E + E + 2×E = 33 which simplifies to 11×E = 33 Divide both sides of the above equation by 11: 11×E ÷ 11 = 33 ÷ 11 which makes E = 3 making A = F = 2×E = 2 × 3 = 6 B = D = 3×E = 3 × 3 = 9 and ABCDEF = 690936