Puzzle for December 31, 2020 ( )
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
Add C and F to both sides of eq.4: B – C + C + F = D – F + C + F which becomes B + F = D + C which may be written as B + F = C + D In eq.2, replace C + D with B + F: E + F = B + F Substract F from both sides of the above equation: E + F – F = B + F – F which makes E = B
Hint #2
eq.6 may be written as: A = (D + E) ÷ 2 Multiply both sides of the above equation by 2: 2 × A = 2 × (D + E) ÷ 2 which becomes eq.6a) 2×A = D + E In eq.3, add A to both sides, and subtract E from both sides: D – A + A – E = A + E + A – E which becomes eq.3a) D – E = 2×A
Hint #3
In eq.6a, substitute D – E for 2×A (from eq.3a): D – E = D + E In the equation above, subtract D from each side, and add E to each side: D – E – D + E = D + E – D + E which makes 0 = 2×E which means 0 = E and also means B = E = 0
Hint #4
Substitute 0 for E in eq.3a: D – 0 = 2×A which makes eq.3b) D = 2×A
Hint #5
Add D, C, and F to both sides of eq.5: C – D + F + D + C + F = A – C + D – F + D + C + F which becomes 2×C + 2×F = A + 2×D Substitute (2×A) for D in the above equation: 2×C + 2×F = A + 2×(2×A) which becomes 2×C + 2×F = A + 4×A which becomes eq.5a) 2×C + 2×F = 5×A
Hint #6
Substitute 0 for E, and 2×A for D in eq.2: 0 + F = C + 2×A which means eq.2a) F = C + 2×A
Hint #7
Substitute (C + 2×A) for F (from eq.2a) in eq.5a: 2×C + 2×(C + 2×A) = 5×A which becomes 2×C + 2×C + 4×A = 5×A which becomes 4×C + 4×A = 5×A Subtract 4×A from each side of the equation above: 4×C + 4×A – 4×A = 5×A – 4×A which makes 4×C = A
Hint #8
Substitute 4×C for A in eq.3b: D = 2(4×C) which makes D = 8×C
Hint #9
Substitute (4×C) for A in eq.2a: F = C + 2×(4×C) which becomes F = C + 8×C which makes F = 9×C
Solution
Substitute 4×C for A, 0 for B and E, 8×C for D, and 9×C for F in eq.1: 4×C + 0 + C + 8×C + 0 + 9×C = 22 which simplifies to 22×C = 22 Divide both sides of the equation above by 22: 22×C ÷ 22 = 22 ÷ 22 which means C = 1 making A = 4×C = 4 × 1 = 4 D = 8×C = 8 × 1 = 8 F = 9×C = 9 × 1 = 9 and ABCDEF = 401809