Puzzle for April 27, 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 non-negative integer.
Scratchpad
Help Area
Hint #1
eq.6 may be written as: (A + B + E) ÷ 3 = F Multiply both sides of the equation above by 3: 3 × (A + B + E) ÷ 3 = 3 × F which becomes eq.6a) A + B + E = 3×F
Hint #2
Add B and F to both sides of eq.4: F – B + B + F = A – F + B + F which becomes eq.4a) 2×F = A + B In eq.6a, replace A + B with 2×F (from eq.4a): 2×F + E = 3×F Subtract 2×F from each side of the equation above: 2×F + E – 2×F = 3×F – 2×F which makes E = F
Hint #3
In eq.3, replace F with E: D + E = C – D + E In the above equation, subtract E from both sides, and add D to both sides: D + E – E + D = C – D + E – E + D which makes 2×D = C
Hint #4
In eq.2, substitute 2×D for C, F for E, and 2×F for A + B (from eq.4a): 2×D + F = 2×F – 2×D In the equation above, subtract F from both sides, and add 2×D to both sides: 2×D + F – F + 2×D = 2×F – 2×D – F + 2×D which makes 4×D = F and also makes E = F = 4×D
Hint #5
Substitute 4×D for E in eq.5: A – 4×D = D + 4×D – A which becomes A – 4×D = 5×D – A Add 4×D and A to both sides of the above equation: A – 4×D + 4×D + A = 5×D – A + 4×D + A which becomes 2×A = 9×D Divide both sides by 2: 2×A ÷ 2 = 9×D ÷ 2 which makes A = 4½×D
Hint #6
Substitute (4×D) for F, and 4½×D for A in eq.4a: 2×(4×D) = 4½×D + B which becomes 8×D = 4½×D + B Subtract 4½×D from each side of the equation above: 8×D – 4½×D = 4½×D + B – 4½×D which makes 3½×D = B
Solution
Substitute 4½×D for A, 3½×D for B, 2×D for C, and 4×D for E and F in eq.1: 4½×D + 3½×D + 2×D + D + 4×D + 4×D = 38 which simplifies to 19×D = 38 Divide both sides of the above equation by 19: 19×D ÷ 19 = 38 ÷ 19 which means D = 2 making A = 4½×D = 4½ × 2 = 9 B = 3½×D = 3½ × 2 = 7 C = 2×D = 2 × 2 = 4 E = F = 4×D = 4 × 2 = 8 and ABCDEF = 974288