Puzzle for June 27, 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 B and D to both sides of eq.3: F - D + B + D = E - B + B + D which becomes F + B = E + D which may be written as F + B = D + E In eq.2, replace D + E with F + B: F + B = A + B Subtract B from both sides of the above equation: F + B - B = A + B - B which makes F = A
Hint #2
In eq.4, replace F with A: C + A = A Subtract A from both sides: C + A - A = A - A which means C = 0
Hint #3
Substitute 0 for C, and A for F in eq.5: A + 0 - D + A = D - E which becomes 2×A - D = D - E In the above equation, add D and E to both sides: 2×A - D + D + E = D - E + D + E which becomes 2×A + E = 2×D Divide both sides by 2: (2×A + E) ÷ 2 = 2×D ÷ 2 which becomes eq.5a) A + ½×E = D
Hint #4
Substitute A + ½×E for D (from eq.5a) in eq.2: A + ½×E + E = A + B which becomes A + 1½×E = A + B Subtract A from each side: A + 1½×E - A = A + B - A which makes 1½×E = B
Hint #5
Substitute 1½×E for B, A + ½×E for D (from eq.5a), and A for F in eq.6: 1½×E + E = A + ½×E + A - 1½×E which becomes 2½×E = 2×A - E Add E to both sides of the above equation: 2½×E + E = 2×A - E + E which means 3½×E = 2×A Divide both sides by 2: 3½×E ÷ 2 = 2×A ÷ 2 which makes 1¾×E = A and which also makes F = A = 1¾×E
Hint #6
Substitute 1¾×E for A in eq.5a: 1¾×E + ½×E = D which makes 2¼×E = D
Solution
Substitute 1¾×E for A and F, 1½×E for B, 0 for C, and 2¼×E for D in eq.1: 1¾×E + 1½×E + 0 + 2¼×E + E + 1¾×E = 33 which simplifies to 8¼×E = 33 Divide both sides by 8¼: 8¼×E ÷ 8¼ = 33 ÷ 8¼ which means E = 4 making A = F = 1¾×E = 1¾ × 4 = 7 B = 1½×E = 1½ × 4 = 6 D = 2¼×E = 2¼ × 4 = 9 and ABCDEF = 760947