Puzzle for September 8, 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
Subtract the left and right sides of eq.5 from the left and right sides of eq.3, respectively: B + E – (A + B) = A + D – (D + F) which becomes B + E – A – B = A + D – D – F which becomes E – A = A – F Add A to both sides of the above equation: E – A + A = A – F + A which becomes eq.3a) E = 2×A – F
Hint #2
In eq.6, replace D with A + F (from eq.2): C + E = A + A + F + F which becomes eq.6a) C + E = 2×A + 2×F
Hint #3
In eq.6a, replace E with 2×A – F (from eq.3a): C + 2×A – F = 2×A + 2×F In the above equation, subtract 2×A from both sides, and add F to both sides: C + 2×A – F – 2×A + F = 2×A + 2×F – 2×A + F which makes C = 3×F
Hint #4
In eq.6, substitute 3×F for C, and B + E for A + D (from eq.3): 3×F + E = B + E + F Subtract F and E from both sides of the equation above: 3×F + E – F – E = B + E + F – F – E which makes 2×F = B
Hint #5
Substitute 3×F for C, and 2×A – F for E (from eq.3a) in eq.4: 3×F + F = A + 2×A – F which becomes 4×F = 3×A – F Add F to both sides of the above equation: 4×F + F = 3×A – F + F which becomes 5×F = 3×A Divide both sides by 3: 5×F ÷ 3 = 3×A ÷ 3 which makes 1⅔×F = A
Hint #6
Substitute 1⅔×F for A in eq.2: D = 1⅔×F + F which makes D = 2⅔×F
Hint #7
Substitute 3×F for C, and 1⅔×F for A in eq.4: 3×F + F = 1⅔×F + E which becomes 4×F = 1⅔×F + E Subtract 1⅔×F from each side of the above equation: 4×F – 1⅔×F = 1⅔×F + E – 1⅔×F which makes 2⅓×F = E
Solution
Substitute 1⅔×F for A, 2×F for B, 3×F for C, 2⅔×F for D, and 2⅓×F for E in eq.1: 1⅔×F + 2×F + 3×F + 2⅔×F + 2⅓×F + F = 38 which simplifies to 12⅔×F = 38 Divide both sides of the above equation by 12⅔: 12⅔×F ÷ 12⅔ = 38 ÷ 12⅔ which means F = 3 making A = 1⅔×F = 1⅔ × 3 = 5 B = 2×F = 2 × 3 = 6 C = 3×F = 3 × 3 = 9 D = 2⅔×F = 2⅔ × 3 = 8 E = 2⅓×F = 2⅓ × 3 = 7 and ABCDEF = 569873