Puzzle for July 22, 2021 ( )
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.4, replace C with E + F (from eq.2): A + D + E = E + F + F which becomes A + D + E = E + 2×F Subtract E from both sides of the above equation: A + D + E – E = E + 2×F – E which becomes eq.4a) A + D = 2×F
Hint #2
eq.3 may be written as: B + F = A + D – B In the above equation, replace A + D with 2×F (from eq.4a): B + F = 2×F – B Subtract F from both sides, and add B to both sides: B + F – F + B = 2×F – B – F + B which makes 2×B = F
Hint #3
In eq.5, substitute 2×B for F: E – 2×B = B – E + 2×B which becomes E – 2×B = 3×B – E Add 2×B and E to both sides of the above equation: E – 2×B + 2×B + E = 3×B – E + 2×B + E which makes 2×E = 5×B Divide both sides by 2: 2×E ÷ 2 = 5×B ÷ 2 which makes E = 2½×B
Hint #4
Substitute 2½×B for E, and 2×B for F in eq.2: C = 2½×B + 2×B which makes C = 4½×B
Hint #5
Substitute (2×B) for F in eq.4a: A + D = 2×(2×B) which becomes eq.4b) A + D = 4×B
Hint #6
Substitute 4½×B for C, and 2½×B for E in eq.6: 4½×B – D + 2½×B = A + B + D which becomes 7×B – D = A + B + D which may be written as eq.6a) 7×B – D = A + D + B
Hint #7
Substitute 4×B for A + D (from eq.4b) in eq.6a: 7×B – D = 4×B + B which becomes 7×B – D = 5×B In the above equation, add D to both sides, and subtract 5×B from both sides: 7×B – D + D – 5×B = 5×B + D – 5×B which makes 2×B = D
Hint #8
Substitute 2×B for D in eq.4b: A + 2×B = 4×B Subtract 2×B from both sides of the equation above: A + 2×B – 2×B = 4×B – 2×B which makes A = 2×B
Solution
Substitute 2×B for A and D and F, 4½×B for C, and 2½×B for E in eq.1: 2×B + B + 4½×B + 2×B + 2½×B + 2×B = 28 which simplifies to 14×B = 28 Divide both sides of the above equation by 14: 14×B ÷ 14 = 28 ÷ 14 which means B = 2 making A = D = F = 2×B = 2 × 2 = 4 C = 4½×B = 4½ × 2 = 9 E = 2½×B = 2½ × 2 = 5 and ABCDEF = 429454