Puzzle for October 2, 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, C, and D to both sides of eq.3: B – A – C + A + C + D = F – D + A + C + D which becomes B + D = F + A + C which is the same as B + D = A + C + F In eq.2, replace A + C + F with B + D: B + D = B + E Subtract B from both sides of the equation above: B + D – B = B + E – B which makes D = E
Hint #2
In eq.5, replace E with D: C + D = A + D Subtract D from both sides of the above equation: C + D – D = A + D – D which makes C = A
Hint #3
In eq.3, subtract B from both sides, and add D to both sides: B – A – C – B + D = F – D – B + D which becomes –A – C + D = F – B In the above equation, substitute C for both A and for F – B (from eq.4): –C – C + D = C which becomes –2×C + D = C Add 2×C to both sides of the equation above: –2×C + D + 2×C = C + 2×C which makes D = 3×C and also makes E = D = 3×C
Hint #4
Substitute C for A, and 3×C for both D and E in eq.6: C + C + 3×C + 3×C = B which makes 8×C = B
Hint #5
Substitute 8×C for B in eq.4: F – 8×C = C Add 8×C to both sides of the above equation: F – 8×C + 8×C = C + 8×C which makes F = 9×C
Solution
Substitute C for A, 8×C for B, 3×C for D and E, and 9×C for F in eq.1: C + 8×C + C + 3×C + 3×C + 9×C = 25 which simplifies to 25×C = 25 ÷ 25 Divide both sides of the above equation by 25: 25×C ÷ 25 = 25 ÷ 25 which means C = 1 making A = C = 1 B = 8×C = 8×1 = 8 D = E = 3×C = 3×1 = 3 F = 9×C = 9×1 = 9 and ABCDEF = 181339