Puzzle for March 11, 2023 ( )
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
Subtract A from both sides of eq.3: D - A = A + C - A which becomes eq.3a) D - A = C
Hint #2
In eq.6, replace C with D - A (from eq.3a): A × D = D - A + D which becomes A × D = 2×D - A In the equation above, subtract (A × D) from both sides, and add A to both sides: A × D - (A × D) + A = 2×D - A - (A × D) + A which becomes A = 2×D - (A × D) which may be written as eq.6a) A = D×(2 - A) Since A and D are non-negative, eq.6a implies: A < 2
Hint #3
Begin confirming: A < 2 If A = 2, then substituting 2 for A in eq.6a would yield: 2 = D×(2 - 2) which would become 2 = D×0 which would make 2 = 0 Since 2 ≠ 0, then: A ≠ 2
Hint #4
Finish confirming: A < 2 If A > 2, then substituting a number > 2 for A (e.g. 3) in eq.6a would yield: 3 = D×(2 - 3) which would become 3 = D×(-1) which would make 3 = -D Multiplying both sides of the above equation by (-1) would yield: (-1) × 3 = (-1) × (-D) which would make -3 = D Since D must be non-negative, then: -3 ≠ D Therefore: A cannot be ≥ 2 which means A < 2 making A = 0 or A = 1
Hint #5
Check: A = 0 If A = 0, then substituting 0 for A in eq.6a would yield: 0 = D×(2 - 0) which would become 0 = D×2 which would make 0 = D which means 0 = D = A
Hint #6
Check: A = 1 If A = 1, then substituting 1 for A in eq.6a would yield: 1 = D×(2 - 1) which would become 1 = D×1 which would make 1 = D which means 1 = D = A So, whether A = 0 or A = 1, then: D = A
Hint #7
Substitute A for D in eq.3a: A - A = C which makes 0 = C
Hint #8
Substitute 0 for C in eq.4: E - 0 = B + 0 which makes E = B
Hint #9
Substitute 0 for C in eq.2: F - 0 = E which makes F = E and also makes F = E = B
Hint #10
Substitute B for F, and 0 for C in eq.5: B ÷ B = A - 0 which makes 1 = A and also makes 1 = A = D
Solution
Substitute 1 for A and D, 0 for C, and B for E and F in eq.1: 1 + B + 0 + 1 + B + B = 23 which becomes 2 + 3×B = 23 Subtract 2 from each side of the equation above: 2 + 3×B - 2 = 23 - 2 which makes 3×B = 21 Divide both sides by 3: 3×B ÷ 3 = 21 ÷ 3 which means B = 7 making E = F = B = 7 and ABCDEF = 170177