Puzzle for August 25, 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
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Hint #1
In eq.5, replace D with A + B (from eq.2): B + F = A + B - B which becomes eq.5a) B + F = A
Hint #2
In eq.4, replace A with B + F (from eq.5a): D - E = B + F - F which becomes D - E = B Add E to both sides of the above equation: D - E + E = B + E which becomes eq.4a) D = B + E
Hint #3
In eq.2, substitute B + E for D (from eq.4a): B + E = A + B Subtract B from each side of the equation above: B + E - B = A + B - B which makes E = A
Hint #4
Substitute B + D for C (from eq.3) in eq.6: E + F - B = B + D + D - F which becomes E + F - B = B + 2×D - F Add B and F to both sides of the above equation: E + F - B + B + F = B + 2×D - F + B + F which becomes eq.6a) E + 2×F = 2×B + 2×D
Hint #5
Subtract B from both sides of eq.4a: D - B = B + E - B which becomes eq.4b) D - B = E
Hint #6
Substitute D - B for E (from eq.4b) into eq.6a: D - B + 2×F = 2×B + 2×D In the above equation, subtract D from both sides, and add B to both sides: D - B + 2×F - D + B = 2×B + 2×D - D + B which becomes eq.6b) 2×F = 3×B + D
Hint #7
Add B to both sides of eq.5: B + F + B = D - B + B which becomes eq.5b) 2×B + F = D
Hint #8
Substitute 2×B + F for D (from eq.5a) into eq.6b: 2×F = 3×B + 2×B + F which becomes 2×F = 5×B + F Subtract F from each side of the above equation: 2×F - F = 5×B + F - F which makes F = 5×B
Hint #9
Substitute 5×B for F in eq.5b: 2×B + 5×B = D which makes 7×B = D
Hint #10
Substitute 7×B for D in eq.4b: 7×B - B = E which makes 6×B = E and also makes 6×B = E = A
Hint #11
Substitute 7×B for D in eq.3: C = B + 7×B which makes C = 8×B
Solution
Substitute 6×B for A and E, 8×B for C, 7×B for D, and 5×B for F in eq.1: 6×B + B + 8×B + 7×B + 6×B + 5×B = 33 which simplifies to 33×B = 33 Divide both sides of the above equation by 33: 33×B ÷ 33 = 33 ÷ 33 which means B = 1 making A = E = 6×B = 6 × 1 = 6 C = 8×B = 8 × 1 = 8 D = 7×B = 7 × 1 = 7 F = 5×B = 5 × 1 = 5 and ABCDEF = 618765