Puzzle for June 16, 2022 ( )
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 and E from both sides of eq.3: A + F – A – E = D + E – A – E which becomes eq.3a) F – E = D – A Add F to both sides of eq.5: E + F + F = A + D – F + F which becomes eq.5a) E + 2×F = A + D
Hint #2
Add the left and right sides of eq.5a to the left and right sides of eq.3a: F – E + E + 2×F = D – A + A + D which makes 3×F = 2×D Divide both sides of the above equation by 2: 3×F ÷ 2 = 2×D ÷ 2 which makes 1½×F = D
Hint #3
In eq.2, replace D with 1½×F: 1½×F – A = A – C Add A to both sides of the above equation: 1½×F – A + A = A – C + A which becomes eq.2a) 1½×F = 2×A – C Add C to both sides of eq.4: A – C + C = C – F + C which becomes eq.4a) A = 2×C – F
Hint #4
In eq.2a, substitute (2×C – F) for A (from eq.4a): 1½×F = 2×(2×C – F) – C which becomes 1½×F = 4×C – 2×F – C which becomes 1½×F = 3×C – 2×F Add 2×F to both sides of the above equation: 1½×F + 2×F = 3×C – 2×F + 2×F which becomes 3½×F = 3×C Divide both sides by 3: 3½×F ÷ 3 = 3×C ÷ 3 which makes 1⅙×F = C
Hint #5
Substitute (1⅙×F) for C in eq.4a: A = 2×(1⅙×F) – F which becomes A = 2⅓×F – F which makes A = 1⅓×F
Hint #6
Substitute 1⅓×F for A, and 1½×F for D in eq.3: 1⅓×F + F = 1½×F + E which becomes 2⅓×F = 1½×F + E Subtract 1½×F from each side of the equation above: 2⅓×F – 1½×F = 1½×F + E – 1½×F which makes ⅚×F = E
Hint #7
Substitute 1⅙×F for C, 1⅓×F for A, and ⅚×F for E in eq.6: 1⅙×F + F = 1⅓×F + B + ⅚×F which becomes 2⅙×F = 2⅙×F + B Subtract 2⅙×F from each side of the above equation: 2⅙×F – 2⅙×F = 2⅙×F + B – 2⅙×F which makes 0 = B
Solution
Substitute 1⅓×F for A, 0 for B, 1⅙×F for C, 1½×F for D, and ⅚×F for E in eq.1: 1⅓×F + 0 + 1⅙×F + 1½×F + ⅚×F + F = 35 which simplifies to 5⅚×F = 35 Divide both sides of the above equation by 5⅚: 5⅚×F ÷ 5⅚ = 35 ÷ 5⅚ which means F = 6 making A = 1⅓×F = 1⅓ × 6 = 8 C = 1⅙×F = 1⅙ × 6 = 7 D = 1½×F = 1½ × 6 = 9 E = ⅚×F = ⅚ × 6 = 5 and ABCDEF = 807956