Puzzle for April 11, 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
Add A to both sides of eq.3: D - A + A = B - E + A which becomes D = A + B - E In eq.5, replace D with A + B - E: A + B - E - F = A - E Subtract (A - E) from both sides of the above equation: A + B - E - F - (A - E) = A - E - (A - E) which is the same as A + B - E - F - A + E = A - E - A + E which simplifies to B - F = 0 Add F to each side: B - F + F = 0 + F which means B = F
Hint #2
In eq.2, replace F with B: B + C = E + B Subtract B from each side of the above equation: B + C - B = E + B - B which makes C = E
Hint #3
Add E to both sides of eq.4: A + C + E + E = D - E + E which becomes A + C + 2×E = D In eq.6, substitute A + C + 2×E for D: A + A + C + 2×E = B + C + E Subtract both C and 2×E from each side of the above equation: A + A + C + 2×E - C - 2×E = B + C + E - C - 2×E which reduces to eq.4a) 2×A = B - E
Hint #4
Substitute D - A for B - E (from eq.3) in eq.4a: 2×A = D - A Add A to each side: 2×A + A = D - A + A which makes eq.6a) 3×A = D
Hint #5
Substitute E for C, and 3×A for D in eq.4: A + E + E = 3×A - E Add (E - A) to both sides of the above equation: A + E + E + (E - A) = 3×A - E + (E - A) which simplifies to 3×E = 2×A Divide both sides by 2: 3×E ÷ 2 = 2×A ÷ 2 which makes 1½×E = A
Hint #6
In eq.6a, substitute (1½×E) for A: 3×(1½×E) = D which becomes 4½×E = D
Hint #7
Substitute 4½×E for D, and 1½×E for A in eq.3: 4½×E - 1½×E = B - E which becomes 3×E = B - E Add E to both sides: 3×E + E = B - E + E which simplifies to 4×E = B and which also makes F = B = 4×E
Solution
Substitute 1½×E for A, 4×E for B and F, E for C, and 4½×E for D in eq.1: 1½×E + 4×E + E + 4½×E + E + 4×E = 32 which simplifies to 16×E = 32 Divide both sides by 16: 16×E ÷ 16 = 32 ÷ 16 which makes E = 2 making A = 1½×E = 1½ × 2 = 3 B = F = 4×E = 4 × 2 = 8 C = E = 2 D = 4½×E = 4½ × 2 = 9 and ABCDEF = 382928