Puzzle for August 30, 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 C with B + D + E (from eq.3): B + D + E + E = A + B + D which becomes B + D + 2×E = A + B + D Subtract B and D from each side of the above equation: B + D + 2×E - B - D = A + B + D - B - D which simplifies to 2×E = A
Hint #2
Add E to both sides of eq.4: B - E + E = E + F + E which becomes B = 2×E + F In the above equation, replace 2×E with A: eq.4a) B = A + F
Hint #3
In eq.4a, substitute D for A + F (from eq.2): B = D
Hint #4
eq.6 may be written as: D = (A + C + E) ÷ 3 Multiply both sides of the above equation by 3: 3 × D = 3 × (A + C + E) ÷ 3 which becomes eq.6a) 3×D = A + C + E
Hint #5
In eq.6a, replace C with B + D + E (from eq.3): 3×D = A + B + D + E + E which becomes 3×D = A + B + D + 2×E Subtract D and 2×E from each side of the equation above: 3×D - D - 2×E = A + B + D + 2×E - D - 2×E which becomes eq.6b) 2×D - 2×E = A + B
Hint #6
Substitute B for D, and 2×E for A in eq.6b: 2×B - 2×E = 2×E + B In the above equation, add 2×E to both sides, and subtract B from both sides: 2×B - 2×E + 2×E - B = 2×E + B + 2×E - B which becomes B = 4×E and also makes B = D = 4×E
Hint #7
Substitute 4×E for B in eq.4: 4×E - E = E + F which becomes 3×E = E + F Subtract E from each side of the above equation: 3×E - E = E + F - E which makes 2×E = F
Hint #8
Substitute 4×E for B and D in eq.3: C = 4×E + 4×E + E which makes C = 9×E
Solution
Substitute 2×E for A and F, 4×E for B and D, and 9×E for C in eq.1: 2×E + 4×E + 9×E + 4×E + E + 2×E = 22 which simplifies to 22×E = 22 Divide both sides of the above equation by 22: 22×E ÷ 22 = 22 ÷ 22 which means E = 1 making A = F = 2×E = 2 × 1 = 2 B = D = 4×E = 4 × 1 = 4 C = 9×E = 9 × 1 = 9 and ABCDEF = 249412