Puzzle for April 1, 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 B + E (from eq.2): A + B = C + B + E + E - A which becomes A + B = C + B + 2×E - A In the above equation, subtract B from both sides, and add A to both sides: A + B - B + A = C + B + 2×E - A - B + A which becomes eq.5a) 2×A = C + 2×E
Hint #2
Add F and E to both sides of eq.4: E - F + F + E = A - E + F + E which becomes eq.4a) 2×E = A + F
Hint #3
In eq.5a, replace 2×E with A + F (from eq.4a): 2×A = C + A + F Subtract A from each side of the equation above: 2×A - A = C + A + F - A which becomes eq.5b) A = C + F
Hint #4
In eq.5b, substitute A + B for C + F (from eq.3): A = A + B Subtract A from each side of the above equation: A - A = A + B - A which makes 0 = B
Hint #5
Substitute 0 for B in eq.2: D = 0 + E which makes D = E
Hint #6
Substitute E for D, C + F for A (from eq.5b), and 0 for B in eq.6: E + E + F = C + F + C - E - F + 0 which becomes 2×E + F = 2×C - E Add E to both sides of the above equation: 2×E + F + E = 2×C - E + E which becomes eq.6a) 3×E + F = 2×C
Hint #7
Substitute C + F for A (from eq.5b) in eq.4a: 2×E = C + F + F which becomes 2×E = C + 2×F Subtract 2×F from both sides of the equation above: 2×E - 2×F = C + 2×F - 2×F which becomes eq.4b) 2×E - 2×F = C
Hint #8
Substitute (2×E - 2×F) for C (from eq.4b) into eq.6a: 3×E + F = 2×(2×E - 2×F) which becomes 3×E + F = 4×E - 4×F In the equation above, subtract 3×E from both sides, and add 4×F to both sides: 3×E + F - 3×E + 4×F = 4×E - 4×F - 3×E + 4×F which makes 5×F = E and also makes 5×F = E = D
Hint #9
Substitute (5×F) for E in eq.4b: 2×(5×F) - 2×F = C which becomes 10×F - 2×F = C which makes 8×F = C
Hint #10
Substitute 8×F for C in eq.5b: A = 8×F + F which makes A = 9×F
Solution
Substitute 9×F for A, 0 for B, 8×F for C, and 5×F for D and E in eq.1: 9×F + 0 + 8×F + 5×F + 5×F + F = 28 which simplifies to 28×F = 28 Divide both sides of the above equation by 28: 28×F ÷ 28 = 28 ÷ 28 which means F = 1 making A = 9×F = 9 × 1 = 9 C = 8×F = 8 × 1 = 8 D = E = 5×F = 5 × 1 = 5 and ABCDEF = 908551