Puzzle for August 18, 2022 ( )
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 to both sides of eq.2: A + F + A = C + E – A + A which becomes eq.2a) 2×A + F = C + E Add C to both sides of eq.4: D + E – A + C = A + B – C – F + C which becomes D + E – A + C = A + B – F which may be written as eq.4a) D + C + E – A = A + B – F
Hint #2
In eq.4a, replace C + E with 2×A + F (from eq.2a): D + 2×A + F – A = A + B – F which becomes D + A + F = A + B – F In the above equation, subtract A from both sides, and add F to both sides: D + A + F – A + F = A + B – F – A + F which becomes eq.4b) D + 2×F = B
Hint #3
In eq.1, substitute D + 2×F for B (from eq.4b): D + 2×F = A + D Subtract D from each side of the equation above: D + 2×F – D = A + D – D which makes 2×F = A
Hint #4
In eq.3, substitute D + 2×F for B (from eq.4b): D + 2×F + C – F = D + F – C which becomes D + F + C = D + F – C In the above equation, subtract D and F from both sides, and add C to both sides: D + F + C – D – F + C = D + F – C – D – F + C which simplifies to 2×C = 0 which means C = 0
Hint #5
Substitute (2×F) for A, and 0 for C in eq.2a: 2×(2×F) + F = 0 + E which becomes 4×F + F = E which makes 5×F = E
Hint #6
Substitute 0 for C, and 5×F for E in eq.5: F = 0 + (D ÷ 5×F) which becomes F = D ÷ 5×F Multiply both sides of the above equation by 5×F: 5×F × F = 5×F × (D ÷ 5×F) which becomes 5×F × F = D which may be written as eq.5a) 5×F² = D
Hint #7
To make eq.5a true, check several possible values for F and D: If F = 0, then D = 5×0² = 5×0 = 0 If F = 1, then D = 5×1² = 5×1 = 5 If F = 2, then D = 5×2² = 5×4 = 20 If F > 2, then D > 20 Since D must be a one-digit integer, and F ≠ 0 (5×F = E, and E ≠ 0 (from eq.5)), then the above equations make: F = 1 and D = 5
Hint #8
Substitute 5 for D, and 1 for F in eq.4b: 5 + 2×1 = B which becomes 5 + 2 = B which makes 7 = B
Solution
Since F = 1, then: A = 2×F = 2 × 1 = 2 E = 5×F = 5 × 1 = 5 and ABCDEF = 270551