Puzzle for September 2, 2021 ( )
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 to both sides of eq.2: E – B + B = B + F + B which becomes eq.2a) E = 2×B + F Add B to both sides of eq.4: C – B + B = A + B + B which becomes eq.4a) C = A + 2×B
Hint #2
In eq.5, replace E with 2×B + F (from eq.2a), and replace C with A + 2×B (from eq.4a): D + 2×B + F = A + A + 2×B + F which becomes D + 2×B + F = 2×A + 2×B + F Subtract 2×B and F from each side of the above equation: D + 2×B + F – 2×B – F = 2×A + 2×B + F – 2×B – F which makes eq.5a) D = 2×A
Hint #3
In eq.3, replace C with A + 2×B (from eq.4a): A + D = A + 2×B + F Subtract A from each side of the equation above: A + D – A = A + 2×B + F – A which becomes eq.3a) D = 2×B + F
Hint #4
In eq.2a, substitute D for 2×B + F (from eq.3a): E = D
Hint #5
Substitute A + 2×B for C (from eq.4a) into eq.6: A + 2×B + E – F = A + F In the equation above, subtract A from both sides, and add F to both sides: A + 2×B + E – F – A + F = A + F – A + F which becomes eq.6a) 2×B + E = 2×F
Hint #6
Substitute 2×B + F for E (from eq.2a) into eq.6a: 2×B + 2×B + F = 2×F which becomes 4×B + F = 2×F Subtract F from each side of the equation above: 4×B + F – F = 2×F – F which makes 4×B = F
Hint #7
Substitute 4×B for F in eq.3a: D = 2×B + 4×B which makes D = 6×B and also makes E = D = 6×B
Hint #8
Substitute 6×B for D in eq.5a: 6×B = 2×A Divide both sides of the above equation by 2: 6×B ÷ 2 = 2×A ÷ 2 which makes 3×B = A
Hint #9
Substitute 3×B for A in eq.4a: C = 3×B + 2×B which makes C = 5×B
Solution
Substitute 3×B for A, 5×B for C, 6×B for D and E, and 4×B for F in eq.1: 3×B + B + 5×B + 6×B + 6×B + 4×B = 25 which simplifies to 25×B = 25 Divide both sides of the above equation by 25: 25×B ÷ 25 = 25 ÷ 25 which means B = 1 making A = 3×B = 3 × 1 = 3 C = 5×B = 5 × 1 = 5 D = E = 6×B = 6 × 1 = 6 F = 4×B = 4 × 1 = 4 and ABCDEF = 315664