Puzzle for June 19, 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 + E (from eq.3): B + B + E = E + F which becomes 2×B + E = E + F Subtract E from each side of the equation above: 2×B + E - E = E + F - E which makes 2×B = F
Hint #2
In eq.2, replace F with 2×B: E = B + 2×B which makes E = 3×B
Hint #3
In eq.3, substitute 3×B for E: C = B + 3×B which makes C = 4×B
Hint #4
Substitute 4×B for C in eq.6: 4×B - A = A - B Add A and B to both sides of the above equation: 4×B - A + A + B = A - B + A + B which becomes 5×B = 2×A Divide both sides by 2: 5×B ÷ 2 = 2×A ÷ 2 which makes 2½×B = A
Hint #5
Substitute 2½×B for A into eq.4: D = 2½×B + B which makes D = 3½×B
Solution
Substitute 2½×B for A, 4×B for C, 3½×B for D, 3×B for E, and 2×B for F into eq.1: 2½×B + B + 4×B + 3½×B + 3×B + 2×B = 32 which simplifies to 16×B = 32 Divide both sides of the above equation by 16: 16×B ÷ 16 = 32 ÷ 16 which means B = 2 making A = 2½×B = 2½ × 2 = 5 C = 4×B = 4 × 2 = 8 D = 3½×B = 3½ × 2 = 7 E = 3×B = 3 × 2 = 6 F = 2×B = 2 × 2 = 4 and ABCDEF = 528764