Puzzle for April 26, 2020 ( )
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.
* AB, BC, and EF are 2-digit numbers (not A×B, B×C, or E×F).
Scratchpad
Help Area
Hint #1
Add B and E to both sides of eq.3: C – B + D – E + B + E = A + B + E + B + E which becomes C + D = A + 2×B + 2×E In the above equation, replace A with B + C (from eq.2): C + D = B + C + 2×B + 2×E Subtract C from each side: C + D – C = B + C + 2×B + 2×E – C which becomes eq.3a) D = 3×B + 2×E
Hint #2
In eq.4, replace D with 3×B + 2×E (from eq.3a): 3×B + 2×E + E = F which becomes eq.4a) 3×B + 3×E = F
Hint #3
eq.6 may be written as: 10×A + B + 10×B + C = D + 10×E + F which becomes 10×A + 11×B + C = D + 10×E + F In the equation above, substitute 3×B + 2×E for D (from eq.3a), and 3×B + 3×E for F (from eq.4a): 10×A + 11×B + C = 3×B + 2×E + 10×E + 3×B + 3×E which becomes 10×A + 11×B + C = 6×B + 15×E Subtract 6×B from each side: 10×A + 11×B + C – 6×B = 6×B + 15×E – 6×B which becomes eq.6a) 10×A + 5×B + C = 15×E
Hint #4
Substitute (B + C) for A (from eq.2) in eq.6a: 10×(B + C) + 5×B + C = 15×E which is equivalent to 10×B + 10×C + 5×B + C = 15×E which becomes eq.6b) 15×B + 11×C = 15×E
Hint #5
Subtract D from both sides of eq.4: D + E – D = F – D which becomes eq.4b) E = F – D In eq.5, add A to both sides, and subtract D from each side: F – A + A – D = A – B + D + A – D which becomes F – D = 2×A – B Substitute E for F – D (from eq.4b) in the above equation: eq.5a) E = 2×A – B
Hint #6
Substitute (B + C) for A (from eq.2) in eq.5a: E = 2×(B + C) – B which is equivalent to E = 2×B + 2×C – B which becomes eq.5b) E = B + 2×C
Hint #7
Substitute (B + 2×C) for E (from eq.5b) in eq.6b: 15×B + 11×C = 15×(B + 2×C) which is equivalent to 15×B + 11×C = 15×B + 30×C Subtract 15×B and 11×C from both sides of the above equation: 15×B + 11×C – 15×B – 11×C = 15×B + 30×C – 15×B – 11×C which simplifies to 0 = 19×C which means 0 = C
Hint #8
Substitute 0 for C in eq.2: B + 0 = A which makes B = A
Hint #9
Substitute 0 for C in eq.5b: E = B + 2×0 which becomes E = B + 0 which means E = B
Hint #10
Substitute B for E in eq.3a: D = 3×B + 2×B which makes D = 5×B
Hint #11
Substitute B for E in eq.4a: 3×B + 3×B = F which makes 6×B = F
Solution
Substitute B for A and E, 0 for C, 5×B for D, and 6×B for F in eq.1: B + B + 0 + 5×B + B + 6×B = 14 which simplifies to 14×B = 14 Divide both sides of the above equation by 14: 14×B ÷ 14 = 14 ÷ 14 which means B = 1 making A = E = B = 1 D = 5×B = 5 × 1 = 5 F = 6×B = 6 × 1 = 6 and ABCDEF = 110516