Puzzle for November 18, 2023 ( )
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
Help Area
Hint #1
eq.6 may be written as: F - B = (B + C + D + E) ÷ 4 Multiply both sides of the above equation by 4: 4 × (F - B) = 4 × (B + C + D + E) ÷ 4 which becomes eq.6a) 4×F - 4×B = B + C + D + E
Hint #2
In eq.4, subtract D from each side, and add A and B to both sides: C + D - A - D + A + B = A - B - E - D + A + B which becomes C + B = 2×A - E - D which may be written as eq.4a) B + C = 2×A - (D + E)
Hint #3
In eq.6a, substitute 2×A - (D + E) for B + C (from eq.4a): 4×F - 4×B = 2×A - (D + E) + D + E which becomes 4×F - 4×B = 2×A Divide both sides of the above equation by 2: (4×F - 4×B) ÷ 2 = 2×A ÷ 2 which becomes eq.6b) 2×F - 2×B = A
Hint #4
In eq.5, replace A with 2×F - 2×B (from eq.6b): 2×F - 2×B + B - F = C + F - B which becomes F - B = C + F - B In the above equation, subtract F from both sides, and add B to both sides: F - B - F + B = C + F - B - F + B which makes 0 = C
Hint #5
In eq.2, substitute 0 for C: B + 0 = D + E - B which becomes B = D + E - B Add B to both sides of the above equation: B + B = D + E - B + B which becomes eq.2b) 2×B = D + E
Hint #6
In eq.4a, replace C with 0, and D + E with 2×B (from eq.2b): B + 0 = 2×A - (2×B) which becomes B = 2×A - 2×B Add 2×B to both sides of the equation above: B + 2×B = 2×A - 2×B + 2×B which makes 3×B = 2×A Divide both sides by 2: 3×B ÷ 2 = 2×A ÷ 2 which makes 1½×B = A
Hint #7
Substitute 1½×B for A in eq.6b: 2×F - 2×B = 1½×B Add 2×B to both sides of the equation above: 2×F - 2×B + 2×B = 1½×B + 2×B which becomes 2×F = 3½×B Divide both sides by 2: 2×F ÷ 2 = 3½×B ÷ 2 which makes F = 1¾×B
Hint #8
Subtract E from both sides of eq.2b: 2×B - E = D + E - E which becomes eq.2c) 2×B - E = D
Hint #9
Substitute 1¾×B for F, 1½×B for A, 0 for C, and 2×B - E for D (from eq.2c) in eq.3: E + 1¾×B = 1½×B + 0 + 2×B - E - 1¾×B which becomes E + 1¾×B = 1¾×B - E In the above equation, subtract 1¾×B from both sides, and add E to both sides: E + 1¾×B - 1¾×B + E = 1¾×B - E - 1¾×B + E which makes 2×E = 0 which means E = 0
Hint #10
Substitute 0 for E in eq.2c: 2×B - 0 = D which makes 2×B = D
Solution
Substitute 1½×B for A, 0 for C and E, 2×B for D, and 1¾×B for F in eq.1: 1½×B + B + 0 + 2×B + 0 + 1¾×B = 25 which simplifies to 6¼×B = 25 Divide both sides of the above equation by 6¼: 6¼×B ÷ 6¼ = 25 ÷ 6¼ which means B = 4 making A = 1½×B = 1½ × 4 = 6 D = 2×B = 2 × 4 = 8 F = 1¾×B = 1¾ × 4 = 7 and ABCDEF = 640807