Puzzle for September 4, 2022 ( )
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
Subtract the left and right sides of eq.5 from the left and right sides of eq.4, respectively: A – B + C – (A + C – F) = B – C + D + E – (B – C + E + F) which becomes A – B + C – A – C + F = B – C + D + E – B + C – E – F which simplifies to –B + F = D – F Add B and F to both sides of the above equation: –B + F + B + F = D – F + B + F which becomes eq.4a) 2×F = D + B
Hint #2
In eq.4a, replace D with E + F (from eq.2): 2×F = E + F + B Subtract F from both sides of the equation above: 2×F – F = E + F + B – F which becomes eq.4b) F = E + B
Hint #3
eq.6 may be written as: F = (A + B + C + E) ÷ 4 Multiply both sides of the above equation by 4: 4 × F = 4 × (A + B + C + E) ÷ 4 which becomes 4×F = A + B + C + E which is the same as eq.6a) 4×F = A + C + E + B
Hint #4
In eq.6a, replace E + B with F (from eq.4b): 4×F = A + C + F Subtract F from each side of the above equation: 4×F – F = A + C + F – F which becomes eq.6b) 3×F = A + C
Hint #5
In eq.3, substitute 3×F for A + C (from eq.6b): D + E = 3×F – D Add D to both sides of the above equation: D + E + D = 3×F – D + D which becomes eq.3a) 2×D + E = 3×F
Hint #6
Subtract E from each side of eq.2: D – E = E + F – E which becomes eq.2a) D – E = F
Hint #7
Substitute (D – E) for F (from eq.2a) into eq.3a: 2×D + E = 3×(D – E) which becomes 2×D + E = 3×D – 3×E In the equation above, subtract 2×D from both sides, and add 3×E to both sides: 2×D + E – 2×D + 3×E = 3×D – 3×E – 2×D + 3×E which simplifies to 4×E = D
Hint #8
Substitute 4×E for D in eq.2a: 4×E – E = F which makes 3×E = F
Hint #9
Substitute 3×E for F in eq.4b: 3×E = E + B Subtract E from both sides of the equation above: 3×E – E = E + B – E which becomes 2×E = B
Hint #10
Substitute (3×E) for F in eq.6b: 3×(3×E) = A + C which becomes eq.6c) 9×E = A + C
Hint #11
Substitute 9×E for A + C (from eq.6c), 3×E for F, and 2×E for B in eq.5: 9×E – 3×E = 2×E – C + E + 3×E which becomes 6×E = 6×E – C Subtract 6×E from each side of the equation above: 6×E – 6×E = 6×E – C – 6×E which makes 0 = –C which means 0 = C
Hint #12
Substitute 0 for C in eq.6c: 9×E = A + 0 which makes 9×E = A
Solution
Substitute 9×E for A, 2×E for B, 0 for C, 4×E for D, and 3×E for F in eq.1: 9×E + 2×E + 0 + 4×E + E + 3×E = 19 which simplifies to 19×E = 19 Divide both sides of the above equation by 19: 19×E ÷ 19 = 19 ÷ 19 which means E = 1 making A = 9×E = 9 × 1 = 9 B = 2×E = 2 × 1 = 2 D = 4×E = 4 × 1 = 4 F = 3×E = 3 × 1 = 3 and ABCDEF = 920413