Puzzle for January 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 positive integer.
Scratchpad
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Hint #1
In eq.4, replace C with A + D (from eq.1): A + D – E = A + B + F In the equation above, subtract A from each side, and add E to each side: A + D – E – A + E = A + B + F – A + E which becomes eq.4a) D = B + F + E
Hint #2
In eq.3, replace D with B + F + E (from eq.4a): B + F + E = A + B + E Subtract B and E from both sides of the equation above: B + F + E – B – E = A + B + E – B – E which simplifies to F = A
Hint #3
In eq.2, substitute A for F: E + A = A + B Subtract A from both sides of the equation above: E + A – A = A + B – A which makes E = B
Hint #4
Substitute B for E in eq.3: D = A + B + B which becomes eq.3a) D = A + 2×B
Hint #5
eq.5 may be written as: B = (A + D + F) ÷ 3 Multiply both sides of the equation above by 3: 3 × B = 3 × (A + D + F) ÷ 3 which becomes eq.5a) 3×B = A + D + F
Hint #6
Substitute A + 2×B for D (from eq.3a), and A for F in eq.5a: 3×B = A + A + 2×B + A which becomes 3×B = 3×A + 2×B Subtract 2×A from each side of the equation above: 3×B – 2×A = 2×A + D – 2×A which becomes eq.5b) 3×B – 2×A = D
Hint #7
Substitute A + 2×B for D (from eq.3a) in eq.5b: 3×B – 2×A = A + 2×B In the equation above, add 2×A to both sides, and subtract 2×B from both sides: 3×B – 2×A + 2×A – 2×B = A + 2×B + 2×A – 2×B which makes B = 3×A and also makes E = B = 3×A
Hint #8
Substitute (3×A) for B in eq.3a: D = A + 2×(3×A) which becomes D = A + 6×A which makes D = 7×A
Hint #9
Substitute 7×A for D in eq.1: A + 7×A = C which makes 8×A = C
Solution
Substitute 8×A for C, A for F, and 3×A for B and E in eq.6: 8×A + A = 3×A × 3×A which becomes 9×A = 9×A² Divide both sides of the equation above by 9×A: 9×A ÷ 9×A = 9×A² ÷ 9×A which means 1 = A making B = E = 3×A = 3×1 = 3 C = 8×A = 8×1 = 8 D = 7×A = 7×1 = 7 F = A = 1 and ABCDEF = 138731