Puzzle for December 9, 2022 ( )
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.
Many thanks to Abby S (age 12) for contributing this fun and interesting puzzle! Thank you, Abby!
Scratchpad
Help Area
Hint #1
In eq.5, replace B + C + D with A + E + F (from eq.2): eq.5a) A × E = A + E + F
Hint #2
In eq.5a, replace E + F with A × B (from eq.3): A × E = A + A × B which may be written as A × E = A × (1 + B) Since A ≠ 0 (from eq.4), divide both sides of the above equation by A: (A × E) ÷ A = (A × (1 + B)) ÷ A which becomes eq.5b) E = 1 + B
Hint #3
In eq.1, substitute A + E + F for B + C + D (from eq.2): A + A + E + F + E + F = 18 which may be written as 2 × (A + E + F) = 18 Divide both sides of the equation above by 2: 2 × (A + E + F) ÷ 2 = 18 ÷ 2 which becomes eq.1a) A + E + F = 9
Hint #4
Substitute A × E for A + E + F (from eq.5a) into eq.1a: A × E = 9 Since A and E are one-digit non-negative integers, the above equation makes either: A = 9 and E = 1 or A = 1 and E = 9 or A = 3 and E = 3
Hint #5
Check: A = 9, and E = 1 ... Substituting 9 for A, and 1 for E in eq.1a would yield: 9 + 1 + F = 9 which would become 10 + F = 9 Subtracting 10 from each side of the equation above would yield: 10 + F - 10 = 9 - 10 which would make F = -1 Since F is non-negative, then: F ≠ -1 which means A ≠ 9 and E ≠ 1
Hint #6
Check: A = 1, and E = 9 ... Substituting 1 for A, and 9 for E in eq.1a would yield: 1 + 9 + F = 9 which would become 10 + F = 9 Subtracting 10 from each side of the equation above would yield: 10 + F - 10 = 9 - 10 which would make F = -1 Since F is non-negative, then: F ≠ -1 which means A ≠ 1 and E ≠ 9 and therefore makes: A = 3 and E = 3
Hint #7
Substitute 3 for E in eq.5b: 3 = 1 + B Subtract 1 from each side of the equation above: 3 - 1 = 1 + B - 1 which makes 2 = B
Hint #8
Substitute 3 for A and E in eq.1a: 3 + 3 + F = 9 which becomes 6 + F = 9 Subtract 6 from each side of the equation above: 6 + F - 6 = 9 - 6 which makes F = 3
Hint #9
Substitute 2 for B, and 3 for A in eq.4: 2 × C = D ÷ 3 Multiply both sides of the above equation by 3: 3 × (2 × C) = 3 × (D ÷ 3) which makes 6×C = D
Hint #10
Substitute 3 for A and E, 2 for B, and 6×C for D in eq.5: 3 × 3 = 2 + C + 6×C which becomes 9 = 2 + 7×C Subtract 2 from both sides of the above equation: 9 - 2 = 2 + 7×C - 2 which makes 7 = 7×C Divide both sides by 7: 7 ÷ 7 = 7×C ÷ 7 which means 1 = C
Solution
Since C = 1, then: D = 6×C = 6 × 1 = 6 and ABCDEF = 321633