Puzzle for April 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
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Hint #1
Add E and F to both sides of eq.4: B - E + E + F = A + E - F + E + F which becomes eq.4a) B + F = A + 2×E
Hint #2
In eq.2, replace B + F with A + 2×E (from eq.4a): A + D = A + 2×E Subtract A from both sides of the above equation: A + D - A = A + 2×E - A which makes D = 2×E
Hint #3
In eq.5, replace D with 2×E, and A - B with C - A (from eq.3): C - 2×E = C - A + 2×E In the above equation, subtract C from both sides, and add 2×E and A to both sides: C - 2×E - C + 2×E + A = C - A + 2×E - C + 2×E + A which simplifies to A = 4×E
Hint #4
In eq.6, substitute 2×E for D, and 4×E for A: 2×E + E = 4×E + B - E which becomes 3×E = 3×E + B Subtract 3×E from each side of the equation above: 3×E - 3×E = 3×E + B - 3×E which makes 0 = B
Hint #5
Substitute 4×E for A, and 0 for B in eq.3: C - 4×E = 4×E - 0 which becomes C - 4×E = 4×E Add 4×E to both sides of the equation above: C - 4×E + 4×E = 4×E + 4×E which makes C = 8×E
Hint #6
Substitute 4×E for A, 2×E for D, and 0 for B in eq.2: 4×E + 2×E = 0 + F which makes 6×E = F
Solution
Substitute 4×E for A, 0 for B, 8×E for C, 2×E for D, and 6×E for F in eq.1: 4×E + 0 + 8×E + 2×E + E + 6×E = 21 which simplifies to 21×E = 21 Divide both sides of the above equation by 21: 21×E ÷ 21 = 21 ÷ 21 which means E = 1 making A = 4×E = 4 × 1 = 4 C = 8×E = 8 × 1 = 8 D = 2×E = 2 × 1 = 2 F = 6×E = 6 × 1 = 6 and ABCDEF = 408216