Puzzle for June 28, 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
Help Area
Hint #1
eq.1 may be written as: A + E + B + C + D + F = 36 In the equation above, replace B + C + D + F with A + E (from eq.5): A + E + A + E = 36 which becomes 2×(A + E) = 36 Divide both sides by 2: 2×(A + E) ÷ 2 = 36 ÷ 2 which becomes eq.1a) A + E = 18
Hint #2
Since A and E are one-digit non-negative integers, then: 0 ≤ A ≤ 9 and 0 ≤ E ≤ 9 making 0 ≤ A + E ≤ 18 Combining all the above inequalities with eq.1a makes: A = 9 and E = 9
Hint #3
In eq.4, replace A with 9: B + C = 9 - B Subtract B from each side of the equation above: B + C - B = 9 - B - B which becomes eq.4a) C = 9 - 2×B
Hint #4
In eq.2, substitute 9 - 2×B for C (from eq.4a): 9 - 2×B = B + D Subtract B from each side of the above equation: 9 - 2×B - B = B + D - B which becomes eq.2a) 9 - 3×B = D
Hint #5
Substitute 9 for A and E, 9 - 2×B for C (from eq.4a), and 9 - 3×B for D (from eq.2a) in eq.5: 9 + 9 = B + 9 - 2×B + 9 - 3×B + F which becomes 18 = 18 - 4×B + F In the above equation, subtract 18 from both sides, and add 4×B to both sides: 18 - 18 + 4×B = 18 - 4×B + F - 18 + 4×B which simplifies to 4×B = F
Hint #6
Substitute 9 for E, 4×B for F, 9 - 2×B for C (from eq.4a), and 9 - 3×B for D (from eq.2a) in eq.3: 9 + 4×B = 9 - 2×B + 9 - 3×B which becomes 9 + 4×B = 18 - 5×B In the above equation, subtract 9 from both sides, and add 5×B to both sides: 9 + 4×B - 9 + 5×B = 18 - 5×B - 9 + 5×B which becomes 9×B = 9 Divide both sides by 9: 9×B ÷ 9 = 9 ÷ 9 which makes B = 1
Solution
Since B = 1, then: C = 9 - 2×B = 9 - 2×1 = 9 - 2 = 7 (from eq.4a) D = 9 - 3×B = 9 - 3×1 = 9 - 3 = 6 (from eq.2a) F = 4×B = 4×1 = 4 and ABCDEF = 917694