Puzzle for April 4, 2021 ( )
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.
Once again, we extend our thanks to Abby S (age 10) for sending us a challenging puzzle! Thank you, Abby!
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Hint #1
Add D to both sides of eq.1: A + D = E + F – D + D which becomes eq.1a) A + D = E + F Add A and F to both sides of eq.4: D + A + F = B + C + E – A – D – F + A + F which becomes D + A + F = B + C + E – D which is the same as eq.4a) A + D + F = B + C + E – D
Hint #2
In eq.4a, replace A + D with E + F (from eq.1a): E + F + F = B + C + E – D which becomes E + 2×F = B + C + E – D Subtract E from each side of the equation above: E + 2×F – E = B + C + E – D – E which becomes eq.4b) 2×F = B + C – D
Hint #3
In eq.4b, replace B with (C × F) – C (from eq.2): 2×F = (C × F) – C + C – D which becomes 2×F = (C × F) – D In the above equation, add D to both sides, and subtract 2×F from both sides: 2×F + D – 2×F = (C × F) – D + D – 2×F which becomes D = (C × F) – 2×F which may be written as eq.4c) D = F × (C – 2)
Hint #4
Add C to both sides of eq.2: B + C = (C × F) – C + C which becomes B + C = (C × F) In eq.6, substitute C × F for B + C: F = (((C × F) × E) – D) ÷ B Multiply both sides by B: B × F = B × (((C × F) × E) – D) ÷ B which becomes B × F = ((C × F) × E) – D which is the same as eq.6a) B × F = (C × E × F) – D
Hint #5
In eq.6a, add D to both sides, and subtract (B × F) from both sides: (B × F) + D – (B × F) = (C × E × F) – D + D – (B × F) which becomes D = (C × E × F) – (B × F) which is equivalent to eq.6b) D = ((C × E) – B) × F
Hint #6
Substitute F × (C – 2) for D (from eq.4c) into eq.6b: F × (C – 2) = ((C × E) – B) × F Since F ≠ 0 (from eq.5), divide both sides of the above equation by F: F × (C – 2) ÷ F = ((C × E) – B) × F ÷ F which becomes eq.6c) C – 2 = (C × E) – B
Hint #7
Multiply both sides of eq.3 by E: E × C = E × ((B + D) ÷ E) which becomes E × C = B + D which may be written as C × E = B + D Substitute B + D for (C × E) in eq.6c: C – 2 = B + D – B which becomes eq.6d) C – 2 = D
Hint #8
In eq.4c, replace (C – 2) with D (from eq.6d): D = F × D The above equation makes F = 1 or D = 0
Hint #9
Check: F = 1 ... Substituting 1 for F in eq.5 would yield: E = ((B × C) + D + E) ÷ 1 which would become E = (B × C) + D + E Subtracting E from each side of the above equation would yield: E – E = (B × C) + D + E – E which would make 0 = (B × C) + D Since B, C, and D are all non-negative integers, the above equation makes: B × C = 0 and D = 0 Therefore, regardless of whether F = 1 or F ≠ 1: D = 0
Hint #10
Substitute 0 for D in eq.6d: C – 2 = 0 Add 2 to both sides of the above equation: C – 2 + 2 = 0 + 2 which becomes C = 2
Hint #11
Substitute 2 for C, and 0 for D in eq.4b: 2×F = B + 2 – 0 which becomes 2×F = B + 2 Subtract 2 from each side of the equation above: 2×F – 2 = B + 2 – 2 which becomes eq.4d) 2×F – 2 = B
Hint #12
Substitute 2×F – 2 for B (from eq.4d), 2 for C, and 0 for D in eq.5: E = (((2×F – 2) × 2) + 0 + E) ÷ F which becomes E = (4×F – 4 + E) ÷ F Multiply both sides of the above equation by F: F × E = F × ((4×F – 4 + E) ÷ F) which becomes F × E = 4×F – 4 + E Subtract E from both sides: F × E – E = 4×F – 4 + E – E which becomes F × E – E = 4×F – 4 which is equivalent to E × (F – 1) = 4×(F – 1) Divide both sides by (F – 1): E × (F – 1) ÷ (F – 1) = 4×(F – 1) ÷ (F – 1) which becomes E = 4 (assumes F ≠ 1)
Hint #13
Confirm: F ≠ 1 ... Substituting 1 for F in eq.2 would yield: B = (C × 1) – C which would become B = C – C which would make B = 0 Since B ≠ 0 (from eq.6), then: F ≠ 1
Hint #14
Substitute 2 for C, 0 for D, and 4 for E in eq.3: 2 = (B + 0) ÷ 4 which becomes 2 = B ÷ 4 Multiply both sides of the above equation by 4: 4 × 2 = 4 × (B ÷ 4) which makes 8 = B
Hint #15
Substitute 8 for B in eq.4d: 2×F – 2 = 8 Add 2 to both sides of the above equation: 2×F – 2 + 2 = 8 + 2 which becomes 2×F = 10 Divide both sides by 2: 2×F ÷ 2 = 10 ÷ 2 which makes F = 5
Solution
Substitute 4 for E, 5 for F, and 0 for D in eq.1: A = 4 + 5 – 0 which makes A = 9 and makes ABCDEF = 982045