Puzzle for November 6, 2021 ( )
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.
* F! means F-factorial.
Scratchpad
Help Area
Hint #1
Replace F on the left side of eq.2 with A – C (from eq.1): C + A – C = B – F which becomes A = B – F Add F to both sides of the above equation: A + F = B – F + F which becomes eq.2a) A + F = B
Hint #2
In eq.3, replace A + F with B (from eq.2a): B + D = B – E In the above equation, subtract B from both sides, and add E to both sides: B + D – B + E = B – E – B + E which simplifies to D + E = 0 Since D and E must be non-negative, the above equation makes: D = 0 and E = 0
Hint #3
In eq.4, substitute 0 for D, and (A + F) for B (from eq.2a): (A ÷ F) – 0 = (A + F) ÷ F – C which is equivalent to A ÷ F = (A ÷ F) + (F ÷ F) – C which becomes A ÷ F = (A ÷ F) + 1 – C In the equation above, subtract (A ÷ F) from both sides, and add C to both sides: A ÷ F – (A ÷ F) + C = (A ÷ F) + 1 – C – (A ÷ F) + C which simplifies to C = 1
Hint #4
Substitute 1 for C in eq.1: F = A – 1 Add 1 to both sides of the equation above: F + 1 = A – 1 + 1 which makes eq.1a) F + 1 = A
Hint #5
Substitute F + 1 for A (from eq.1a) into eq.2a: F + 1 + F = B which becomes eq.2b) 2×F + 1 = B
Hint #6
Substitute 2×F + 1 for B (from eq.2b), 1 for C, and 0 for E in eq.5: F! = 2×F + 1 – 1 + 0 which simplifies to F! = 2×F which may be written as F × (F – 1)! = 2×F Since F ≠ 0 (from eq.4), divide both sides of the above equation by F: F × (F – 1)! ÷ F = 2×F ÷ F which becomes (F – 1)! = 2 which makes eq.5a) F – 1 = 2
Solution
Add 1 to both sides of eq.5a: F – 1 + 1 = 2 + 1 which means F = 3 making A = F + 1 = 3 + 1 = 4 (from eq.1a) B = 2×F + 1 = 2×3 + 1 = 6 + 1 = 7 (from eq.2b) and ABCDEF = 471003