Puzzle for March 1, 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.
Scratchpad
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Hint #1
Add D and A to both sides of eq.2: A – D + D + A = B – A + D + A which becomes eq.2a) 2×A = B + D In eq.5, replace B + D with 2×A (from eq.2a): 2×A = A × F Since A ≠ 0 (from eq.6), divide both sides of the above equation by A: 2×A ÷ A = (A × F) ÷ A which makes 2 = F
Hint #2
In eq.4, replace F with 2: 2 = A ÷ D Multiply both sides of the above equation by D: 2 × D = (A ÷ D) × D which makes 2×D = A
Hint #3
In eq.2a, substitute (2×D) for A: 2×(2×D) = B + D which becomes 4×D = B + D Subtract D from each side of the equation above: 4×D – D = B + D – D which makes 3×D = B
Hint #4
Substitute 3×D for B, 2 for F, and 2×D for A in eq.6: D = (3×D × 2) ÷ 2×D which becomes D = 6×D ÷ 2×D which makes D = 3 making A = 2×D = 2 × 3 = 6 B = 3×D = 3 × 3 = 9
Hint #5
Substitute 9 for B, and 2 for F in eq.1: 9 = C + 2 Subtract 2 from each side of the above equation: 9 – 2 = C + 2 – 2 which makes 7 = C
Solution
Substitute 7 for C, 2 for F, 6 for A, and 3 for D in eq.3: 7 – E + 2 = 6 + 3 + E which becomes 9 – E = 9 + E In the equation above, subtract 9 from both sides, and add E to both sides: 9 – E – 9 + E = 9 + E – 9 + E which makes 0 = 2×E which means 0 = E and makes ABCDEF = 697302