Puzzle for February 3, 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
Add B and E to both sides of eq.2: A - B + B + E = F - E + B + E which becomes eq.2a) A + E = F + B Add A to both sides of eq.3: D - A + A = A + E + A which becomes eq.3a) D = A + E + A
Hint #2
In eq.3a, replace A + E with F + B (from eq.2a): D = F + B + A which may be written as eq.3b) D = A + B + F
Hint #3
Add B and F to both sides of eq.4: F - B + B + F = A - F + B + F which becomes eq.4a) 2×F = A + B
Hint #4
In eq.3b, replace A + B with 2×F (from eq.4a): D = 2×F + F which makes eq.3c) D = 3×F
Hint #5
In eq.6, substitute 3×F for D, and 2×F for A + B (from eq.4a): C + 3×F = 2×F - C + F which becomes C + 3×F = 3×F - C In the above equation, subtract 3×F from both sides, and add C to both sides: C + 3×F - 3×F + C = 3×F - C - 3×F + C which makes 2×C = 0 which means C = 0
Hint #6
Substitute 0 for C in eq.5: E - 0 = B - E which becomes E = B - E Add E to both sides of the above equation: E + E = B - E + E which makes 2×E = B
Hint #7
Substitute 2×E for B in eq.2: A - 2×E = F - E Add 2×E to both sides of the above equation: A - 2×E + 2×E = F - E + 2×E which becomes eq.2b) A = F + E
Hint #8
Substitute 2×E for B, and F + E for A (from eq.2b) in eq.4: F - 2×E = F + E - F which becomes F - 2×E = E Add 2×E to both sides of the equation above: F - 2×E + 2×E = E + 2×E which makes F = 3×E
Hint #9
Substitute 3×E for F in eq.2b: A = 3×E + E which makes A = 4×E
Hint #10
Substitute (3×E) for F in eq.3c: D = 3×(3×E) which makes D = 9×E
Solution
Substitute 4×E for A, 2×E for B, 0 for C, 9×E for D, and 3×E for F in eq.1: 4×E + 2×E + 0 + 9×E + E + 3×E = 19 which simplifies to 19×E = 19 Divide both sides of the above equation by 19: 19×E ÷ 19 = 19 ÷ 19 which means E = 1 making A = 4×E = 4 × 1 = 4 B = 2×E = 2 × 1 = 2 D = 9×E = 9 × 1 = 9 F = 3×E = 3 × 1 = 3 and ABCDEF = 420913