Puzzle for August 28, 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.
* BC, CD, and AB are 2-digit numbers (not B×C, C×D, or A×B).
Scratchpad
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Hint #1
Subtract B from both sides of eq.3: B + E – B = A + D – E – B which becomes E = A + D – E – B which may be written as eq.3a) E = A – B + D – E Subtract B and E from both sides of eq.4: A + E – B – E = B + D + F – B – E which becomes eq.4a) A – B = D + F – E
Hint #2
In eq.3a, replace A – B with D + F – E (from eq.4a): E = D + F – E + D – E which becomes E = 2×D + F – 2×E Add 2×E to both sides of the above equation: E + 2×E = 2×D + F – 2×E + 2×E which becomes eq.4b) 3×E = 2×D + F
Hint #3
eq.6 may be written as: F = (D + E) ÷ 2 Multiply both sides of the above equation by 2: F × 2 = ((D + E) ÷ 2) × 2 which becomes 2×F = D + E Subtract D from both sides: 2×F – D = D + E – D which becomes eq.6a) 2×F – D = E
Hint #4
In eq.4b, substitute (2×F – D) for E (from eq.6a): 3×(2×F – D) = 2×D + F which becomes 6×F – 3×D = 2×D + F In the equation above, add 3×D to both sides, and subtract F from both sides: 6×F – 3×D + 3×D – F = 2×D + F + 3×D – F which becomes 5×F = 5×D Divide both sides by 5: 5×F ÷ 5 = 5×D ÷ 5 which makes F = D
Hint #5
In eq.6a, replace F with D: 2×D – D = E which makes D = E
Hint #6
In eq.3a, substitute D for E: D = A – B + D – D which becomes eq.3b) D = A – B
Hint #7
Substitute A – B for D (from eq.3b) in eq.2: C + A – B = A + B In the above equation, subtract A from both sides, and add B to both sides: C + A – B – A + B = A + B – A + B which makes C = 2×B
Hint #8
eq.5 may be written as: 10×B + C + 10×C + D = 10×A + B – D – E – F which becomes 10×B + 11×C + D = 10×A + B – D – E – F Substitute (2×B) for C, and D for E and F in the above equation: 10×B + 11×(2×B) + D = 10×A + B – D – D – D which becomes 10×B + 22×B + D = 10×A + B – 3×D which becomes eq.5a) 32×B + D = 10×A + B – 3×D
Hint #9
Substitute (A – B) for D (from eq.3b) in eq.5a: 32×B + (A – B) = 10×A + B – 3×(A – B) which becomes 32×B + A – B = 10×A + B – 3×A + 3×B which becomes 31×B + A = 7×A + 4×B Subtract A and 4×B from both sides of the above equation: 31×B + A – A – 4×B = 7×A + 4×B – A – 4×B which makes 27×B = 6×A Divide both sides by 6: 27×B ÷ 6 = 6×A ÷ 6 which makes 4½×B = A
Hint #10
Substitute 4½×B for A in eq.3b: D = 4½×B – B which makes D = 3½×B and also makes F = D = E = 3½×B
Solution
Substitute 4½×B for A, 2×B for C, and 3½×B for D and E and F in eq.1: 4½×B + B + 2×B + 3½×B + 3½×B + 3½×B = 36 which simplifies to 18×B = 36 Divide both sides of the above equation by 18: 18×B ÷ 18 = 36 ÷ 18 which means B = 2 making A = 4½×B = 4½ × 2 = 9 C = 2×B = 2 × 2 = 4 D = E = F = 3½×B = 3½ × 2 = 7 and ABCDEF = 924777