Puzzle for November 27, 2019 ( )
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 A and E to both sides of eq.5: D – A + A + E = B – E + A + E which becomes D + E = B + A which may be written as D + E = A + B In eq.2, replace A + B with D + E: D + E = C + E Subtract E from both sides of the above equation: D + E – E = C + E – E which makes D = C
Hint #2
In eq.5, add E to both sides, and subtract D from both sides: D – A + E – D = B – E + E – D which becomes –A + E = B – D which may be written as E – A = B – D In eq.4, replace B – D with E – A: E – A = E – F Subtract E from both sides of the above equation: E – A – E = E – F – E which makes –A = –F which means A = F
Hint #3
Substitute C for D, and A for F in eq.6: C + C – A = A + E which becomes 2×C – A = A + E Subtract A from both sides of the above equation: 2×C – A – A = A + E – A which becomes eq.6a) 2×C – 2×A = E
Hint #4
Substitute 2×C – 2×A for E (from eq.6a), C for D, and A for F in eq.3: 2×C – 2×A – A = C + A which becomes 2×C – 3×A = C + A In the above equation, add 3×A to each side, and subtract C from both sides: 2×C – 3×A + 3×A – C = C + A + 3×A – C which makes C = 4×A and also makes D = C = 4×A
Hint #5
Substitute (4×A) for C in eq.6a: 2×(4×A) – 2×A = E which becomes 8×A – 2×A = E which makes 6×A = E
Hint #6
Substitute 4×A for C, and 6×A for E in eq.2: A + B = 4×A + 6×A which becomes A + B = 10×A Subtract A from both sides of the equation above: A + B – A = 10×A – A which makes B = 9×A
Solution
Substitute 9×A for B, 4×A for C and D, 6×A for E, and A for F in eq.1: A + 9×A + 4×A + 4×A + 6×A + A = 25 which simplifies to 25×A = 25 Divide both sides of the equation above by 25: 25×A ÷ 25 = 25 ÷ 25 which means A = 1 making B = 9×A = 9 × 1 = 9 C = D = 4×A = 4 × 1 = 4 E = 6×A = 6 × 1 = 6 F = A = 1 and ABCDEF = 194461