Puzzle for April 15, 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.
Scratchpad
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Hint #1
In eq.6, subtract C from both sides, and add F to each side: A – F – C + F = C – D – C + F which becomes A – C = –D + F In eq.2, replace A – C with –D + F: –D + F = D + F Subtract F from both sides, and add D to each side: –D + F – F + D = D + F – F + D which makes 0 = 2×D which means 0 = D
Hint #2
In eq.3, replace D with 0: 0 + E = A – E which becomes E = A – E Add E to each side of the equation above: E + E = A – E + E which makes 2×E = A
Hint #3
eq.5 may be written as: B – A = C – B – E Add the left and right sides of the equation above to the left and right sides of eq.4, respectively: A – B + B – A = B + C + E + C – B – E which simplifies to 0 = 2×C which means 0 = C
Hint #4
Substitute 0 for C and D in eq.2: A – 0 = 0 + F which makes A = F and also makes eq.2a) F = A = 2×E
Hint #5
Substitute 2×E for A, and 0 for C in eq.4: 2×E – B = B + 0 + E which becomes 2×E – B = B + E In the equation above, add B to each side, and subtract E from each side: 2×E – B + B – E = B + E + B – E which makes E = 2×B
Hint #6
Substitute (2×B) for E in eq.2a: F = A = 2×(2×B) which makes F = A = 4×B
Solution
Substitute 4×B for A and F, 0 for C and D, and 2×B for E in eq.1: 4×B + B + 0 + 0 + 2×B + 4×B = 11 which simplifies to 11×B = 11 Divide both sides of the equation above by 11: 11×B ÷ 11 = 11 ÷ 11 which means B = 1 making A = F = 4×B = 4 × 1 = 4 E = 2×B = 2 × 1 = 2 and ABCDEF = 410024