Puzzle for December 13, 2018 ( )
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
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Hint #1
Substitute B + D for A (from eq.3) in eq.5: B + D + B + E = C + D Subtract D from each side of the above equation: B + D + B + E - D = C + D - D which becomes 2×B + E = C Substitute 2×B + E for C in eq.4: 2×B + E = E + F Subtract E from each side: 2×B + E - E = E + F - E which makes eq.4a) 2×B = F
Hint #2
In eq.2, replace C with E + F (from eq.4): E + F + E = D + F Subtract F from each side: E + F + E - F = D + F - F which means eq.2a) 2×E = D
Hint #3
Substitute 2×E for D in eq.6: 2×E = E × F To make the above equation true, then: E = 0 and / or: F = 2
Hint #4
Check: E = 0 ... In eq.2a, replace E with 0: 2×0 = D which would make 0 = D Substitute 0 for D in eq.3: A = B + 0 which would mean A = B
Hint #5
Continue checking: E = 0 ... In eq.2, replace D and E with 0: C + 0 = 0 + F which would make C = F Substitute A for B, and 0 for D and E in eq.5: A + A + 0 = C + 0 which would mean 2×A = C
Hint #6
Finish checking: E = 0 ... Substitute A for B, 2×A for C and F, and 0 for D and E in eq.1: A + A + 2×A + 0 + 0 + 2×A = 30 which would simplify to 6×A = 30 Dividing both sides by 6 would yield: 6×A ÷ 6 = 30 ÷ 6 which would mean A = 5 and would make C = 2×A = 2 × 5 = 10 However, C must be a one-digit non-negative integer, which means C ≠ 10 and means E ≠ 0 and therefore makes F = 2
Hint #7
Replace F with 2 in eq.4a: 2×B = 2 Divide both sides of the above equation by 2: 2×B ÷ 2 = 2 ÷ 2 which makes B = 1
Hint #8
Replace F with 2 in eq.4: eq.4b) C = E + 2
Hint #9
Substitute 1 for B, and 2×E for D in eq.3: eq.3a) A = 1 + 2×E
Solution
Substitute 1 + 2×E for A (from eq.3a), 1 for B, E + 2 for C (from eq.4b), 2×E for D, and 2 for F in eq.1: 1 + 2×E + 1 + E + 2 + 2×E + E + 2 = 30 which simplifies to 6×E + 6 = 30 Subtract 6 from each side: 6×E + 6 - 6 = 30 - 6 which means 6×E = 24 Divide each side by 6: 6×E ÷ 6 = 24 ÷ 6 which makes E = 4 making A = 1 + 2×E = 1 + 2×4 = 1 + 8 = 9 (from eq.3a) C = E + 2 = 4 + 2 = 6 (from eq.4b) D = 2×E = 2×4 = 8 (from eq.2a) and ABCDEF = 916842