Puzzle for May 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.3, replace D with A + B (from eq.2): A + B + F = B + C Subtract B from both sides of the above equation: A + B + F - B = B + C - B which becomes A + F = C Replace A + F with B (from eq.5) in the equation above: B = C
Hint #2
In eq.4, add F to each side, and subtract E from each side: C + E + F - E = D - F + F - E which becomes C + F = D - E In eq.6, replace C + F with D - E: A + D + E = B + D - E In the above equation, subtract D from each side, and add E to each side: A + D + E - D + E = B + D - E - D + E which becomes eq.6a) A + 2×E = B
Hint #3
In eq.5, substitute A + 2×E for B (from eq.6a): A + 2×E = A + F Subtract A from each side of the above equation: A + 2×E - A = A + F - A which becomes 2×E = F
Hint #4
Substitute 2×E for F in eq.4: C + E = D - 2×E Add 2×E to both sides of the equation above: C + E + 2×E = D - 2×E + 2×E which becomes eq.4a) C + 3×E = D
Hint #5
Substitute C + 3×E for D (from eq.4a), 2×E for F, and C for B in eq.3: C + 3×E + 2×E = C + C which becomes C + 5×E = 2×C Subtract C from both sides: C + 5×E - C = 2×C - C which makes 5×E = C which also makes B = C = 5×E
Hint #6
Substitute 5×E for B in eq.6a: A + 2×E = 5×E Subtract 2×E from both sides: A + 2×E - 2×E = 5×E - 2×E which means A = 3×E
Hint #7
Substitute 5×E for C in eq.4a: 5×E + 3×E = D which means 8×E = D
Solution
Substitute 3×E for A, 5×E for B and C, 8×E for D, and 2×E for F in eq.1: 3×E + 5×E + 5×E + 8×E + E + 2×E = 24 which simplifies to 24×E = 24 Divide both sides of the above equation by 24: 24×E ÷ 24 = 24 ÷ 24 which means E = 1 making A = 3×E = 3 × 1 = 3 B = C = 5×E = 5 × 1 = 5 D = 8×E = 8 × 1 = 8 F = 2×E = 2 × 1 = 2 and ABCDEF = 355812