Puzzle for November 22, 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 C, E, and F to both sides of eq.6: C – F + C + E + F = A – C – E + C + E + F which becomes eq.6a) 2×C + E = A + F In eq.2, replace A + F with 2×C + E (from eq.6a): C + E = 2×C + E Subtract C and E from both sides of the above equation: C + E – C – E = 2×C + E – C – E which makes 0 = C
Hint #2
Add B and E to both sides of eq.3: A – B + B + E = D – E + B + E which becomes eq.3a) A + E = D + B eq.4 may be written as: C + D + B = E + F In the equation above, replace C with 0, and replace D + B with A + E (from eq.3a): 0 + A + E = E + F Subtract E from both sides of the equation above: 0 + A + E – E = E + F – E which makes eq.4a) A = F
Hint #3
Substitute 0 for C, and A for F in eq.6a: 2×0 + E = A + A which makes E = 2×A
Hint #4
Substitute 2×A for E in eq.3a: A + 2×A = D + B which becomes eq.3b) 3×A = D + B
Hint #5
In eq.5, subtract B from both sides, and add D to each side: D – B – B + D = A + B – D + E – B + D which becomes 2×D – 2×B = A + E Substitute 2×A for E in the equation above: 2×D – 2×B = A + 2×A which becomes eq.5a) 2×D – 2×B = 3×A
Hint #6
Substitute D + B for 3×A (from eq.3b) in eq.5a: 2×D – 2×B = D + B In the above equation, add 2×B to both sides, and subtract D from both sides: 2×D – 2×B + 2×B – D = D + B + 2×B – D which makes eq.5b) D = 3×B
Hint #7
Substitute 3×B for D in eq.3b: 3×A = 3×B + B which makes 3×A = 4×B Divide both sides of the above equation by 4: 3×A ÷ 4 = 4×B ÷ 4 which makes ¾×A = B
Hint #8
Substitute (¾×A) for B in eq.5b: D = 3×(¾×A) which makes D = 2¼×A
Solution
Substitute ¾×A for B, 0 for C, 2¼×A for D, 2×A for E, and A for F in eq.1: A + ¾×A + 0 + 2¼×A + 2×A + A = 28 which simplifies to 7×A = 28 Divide both sides of the equation above by 7: 7×A ÷ 7 = 28 ÷ 7 which means A = 4 making B = ¾×A = ¾ × 4 = 3 D = 2¼×A = 2¼ × A = 9 E = 2×A = 2 × 4 = 8 F = A = 4 and ABCDEF = 430984