Puzzle for June 21, 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 is a 2-digit number (not A×B).
Scratchpad
Help Area
Hint #1
In eq.3, substitute (C – D) for A (from eq.4): E – (C – D) = F which is the same as E – C + D = F Subtract D from each side of the above equation: E – C + D – D = F – D which becomes eq.3a) E – C = F – D
Hint #2
Add the left and right sides of eq.2 to the left and right sides of eq.3a, respectively: E – C + C – B = F – D + D – F which simplifies to E – B = 0 Add B to both sides of the equation above: E – B + B = 0 + B which makes E = B
Hint #3
In eq.5, replace E with B: D – B + B = C – B which becomes eq.5a) D = C – B
Hint #4
In eq.4, substitute (C – B) for D (from eq.5a): C – (C – B) = A which is the same as C – C + B = A which makes B = A
Hint #5
Substitute B for both E and A in eq.3: B – B = F which makes 0 = F
Hint #6
eq.6 may be written as: 10×A + B = (C + D) × E Substitute B for both A and E in the above equation: 10×B + B = (C + D) × B which becomes 11×B = (C + D) × B Subtract 11×B from each side: 11×B – 11×B = (C + D) × B – 11×B which may be written as 0 = (C + D – 11) × B making 0 = C + D – 11 or 0 = B
Hint #7
Check: B = 0 ... Substituting 0 for B, A, E, and F in eq.1 would yield: 0 + 0 + C + D + 0 + 0 = 26 which would make C + D = 26 However, C and D must be one-digit non-negative integers, which means: C ≤ 9 and D ≤ 9 and therefore means C + D ≤ 18 Consequently, C + D ≠ 26 which means B ≠ 0 and therefore means eq.1a) 0 = C + D – 11
Hint #8
Add 11 to both sides of eq.1a: 0 + 11 = C + D – 11 + 11 which means 11 = C + D Substitute B for A and E, 11 for C + D, and 0 for F in eq.1: B + B + 11 + B + 0 = 26 which becomes 3×B + 11 = 26 Subtract 11 from both sides of the equation above: 3×B + 11 – 11 = 26 – 11 which means 3×B = 15 Divide each side by 3: 3×B ÷ 3 = 15 ÷ 3 which makes B = 5 and also makes A = E = B = 5
Hint #9
Substitute 5 for A in eq.4: C – D = 5 Add D to each side of the equation above: C – D + D = 5 + D which makes eq.4a) C = 5 + D
Solution
Substitute 5 + D for C (from eq.4a) in eq.1a: 0 = 5 + D + D – 11 which becomes 0 = 2×D – 6 Add 6 to each side of the above equation: 0 + 6 = 2×D – 6 + 6 which means 6 = 2×D Divide each side by 2: 6 ÷ 2 = 2×D ÷ 2 which makes 3 = D making C = 5 + D = 5 + 3 = 8 (from eq.4a) and ABCDEF = 558350