Puzzle for July 25, 2023 ( )
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.5, replace D with B + E (from eq.4): C + E = B + B + E which becomes C + E = 2×B + E Subtract E from each side of the equation above: C + E - E = 2×B + E - E which makes C = 2×B
Hint #2
In eq.4, replace E with A + B (from eq.3): D = B + A + B which becomes eq.4a) D = 2×B + A
Hint #3
In eq.2, substitute 2×B + A for D (from eq.4a): 2×B + A = A + F Subtract A from both sides of the equation above: 2×B + A - A = A + F - A which makes 2×B = F
Hint #4
eq.6 may be written as: E = (A + B + C + F) ÷ 4 Multiply both sides of the above equation by 4: 4 × E = 4 × (A + B + C + F) ÷ 4 which becomes eq.6a) 4×E = A + B + C + F
Hint #5
Substitute (A + B) for E (from eq.3), and 2×B for C and F in eq.6a: 4×(A + B) = B + 2×B + A + 2×B which becomes 4×A + 4×B = 5×B + A Subtract 4×B and A from each side of the above equation: 4×A + 4×B - 4×B - A = 5×B + A - 4×B - A which makes 3×A = B
Hint #6
Since B = 3×A, then: C = 2×B = 2×(3×A) = 6×A and F = 2×B = 2×(3×A) = 6×A
Hint #7
Substitute 3×A for B in eq.3: E = A + 3×A which makes E = 4×A
Hint #8
Substitute (3×A) for B in eq.4a: D = 2×(3×A) + A which becomes D = 6×A + A which makes D = 7×A
Solution
Substitute 3×A for B, 6×A for C and F, 7×A for D, and 4×A for E in eq.1: A + 3×A + 6×A + 7×A + 4×A + 6×A = 27 which simplifies to 27×A = 27 Divide both sides of the above equation by 27: 27×A ÷ 27 = 27 ÷ 27 which means A = 1 making B = 3×A = 3 × 1 = 3 C = F = 6×A = 6 × 1 = 6 D = 7×A = 7 × 1 = 7 E = 4×A = 4 × 1 = 4 and ABCDEF = 136746