Puzzle for January 19, 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.3, substitute (B + F) for D (from eq.4): A – B = C – (B + F) which is equivalent to A – B = C – B – F Add B and F to both sides of the above equation: A – B + B + F = C – B – F + B + F which becomes eq.3a) A + F = C
Hint #2
In eq.6, substitute A + F for C (from eq.3a): A + F + F = A + E which becomes A + 2×F = A + E Subtract A from both sides of the equation above: A + 2×F – A = A + E – A which makes 2×F = E
Hint #3
In eq.2, replace E with 2×F: 2×F + F = B which makes 3×F = B
Hint #4
In eq.4, replace B with 3×F: D = 3×F + F which makes D = 4×F
Hint #5
In eq.5, substitute 3×F for B, 4×F for D, and A + F for C (from eq.3a): 3×F + 4×F + F = A + A + F which becomes 8×F = 2×A + F Subtract F from both sides of the equation above: 8×F – F = 2×A + F – F which makes 7×F = 2×A Divide both sides by 2: 7×F ÷ 2 = 2×A ÷ 2 which makes 3½×F = A
Hint #6
Substitute 3½×F for A in eq.3a: 3½×F + F = C which makes 4½×F = C
Solution
Substitute 3½×F for A, 3×F for B, 4½×F for C, 4×F for D, and 2×F for E in eq.1: 3½×F + 3×F + 4½×F + 4×F + 2×F + F = 36 which simplifies to 18×F = 36 Divide both sides of the equation above by 18: 18×F ÷ 18 = 36 ÷ 18 which means F = 2 making A = 3½×F = 3½ × 2 = 7 B = 3×F = 3 × 2 = 6 C = 4½×F = 4½ × 2 = 9 D = 4×F = 4 × 2 = 8 E = 2×F = 2 × 2 = 4 and ABCDEF = 769842