1. Complete the sequence described with 3x3 grids and patterns. Without specific visuals or pattern descriptions, this part cannot be solved precisely from the text alone. 2. Shapes with arrows and shaded squares are also missing exact descriptions for logical deduction; cannot determine answers without images or pattern rules. 3. For analogy "X is to Y as Z is to ?": X: circle half shaded Y: circle quarter shaded Z: pentagon with one triangle shaded The analogy reduces shading by half from X to Y (1/2 to 1/4), so from pentagon (Z) shaded 1 part, answer should be pentagon with half that shaded (1/2 of one triangle) or approximately one eighth shaded. Without exact options, cannot finalize. 4. Sequence: 2020, 2062, 2019, x, 2018, 2060 Step 1. Observe pattern between the two interleaved sequences. Odd positions: 2020, 2019, 2018 (decreasing by 1) Even positions: 2062, x, 2060 Even positions decrease by 2: 2062 to x, then x to 2060 So, x = 2062 - 2 = 2060 (but 2060 is last number); so between 2062 and 2060, x must be 2061 to continue decrement by 1? Checking options: 2063, 2064, 2059, 2061 So x = 2061 to fit decrement by 1 in even positions. Answer: d.) 2061 5. Sequence: 189, 199, 211, 225, 241, x Step 1. Find differences: 199 - 189 = 10 211 - 199 = 12 225 - 211 = 14 241 - 225 = 16 Pattern: Differences increase by 2 Next difference = 18 x = 241 + 18 = 259 Answer: a.) 259 Part 2. 6 - 7.) Given 4 rows with numbers and missing ? in row 4, question is the missing number? Format unclear; likely sums or pattern using rows. Rows: 7 5 26 3 2 1 5 3 8 6 44 2 4 9 ? 5 Investigate relationship between numbers in each group: First line: 7, 5 Second line: 26 Third line: 3 Try to relate 7,5 with 26 and 3: 7 * 5 = 35; 35 - 9 = 26; 9 = 3 * 3? Check next group: 2 * 1 = 2; but middle number is 5 This doesn't fit. Try sum first row * third row = middle number? 7 + 5 =12; 12 * 3 = 36 (no match) Next group: 2 + 1 =3; 3 * 3=9 (no match, middle 5) Next: 8 + 6=14; 14*2=28 (middle 44) No clear pattern. Try sum of squares: 7^2 + 5^2=49 + 25=74; No match. Try difference of squares: 7^2 - 5^2=49 - 25=24 (near 26) Try sum of squares + third number: 74 + 3=77 (no match) Try multiplying first and third rows: 7 * 3=21 near 26 2 * 3=6 near 5 8 * 2=16 near 44 no Insufficient info to determine ? 8 - 9.) Circles divided into three parts with numbers. Find missing number in third circle (56,16,??) Check sum of sections for each circle: First: 56 + 27 + 79 = 162 Second: 88 + 23 + 27 = 138 Third: 56 + 16 + x Look for pattern in sums? Or differences? Difference between first and second total: 162 - 138 = 24 If each decreases by 24, third total = 138 - 24 = 114 Calculate missing x: 56 + 16 + x =114 x=114 - 72 = 42 Answer: 42 Part 3. Magic square: Given these cells: Top-right = 7 Middle-left = 13 Middle-center = 37 All rows, columns, and diagonals sum to 111. Let the magic square be: $$\begin{matrix} a & b & 7 \\ 13 & 37 & c \\ d & e & f \end{matrix}$$ Step 1. Use row sums: Row 1: a + b + 7 = 111 \implies a + b = 104 Row 2: 13 + 37 + c = 111 \implies c = 61 Row 3: d + e + f = 111 Step 2. Column sums: Col 1: a + 13 + d = 111 \implies a + d = 98 Col 2: b + 37 + e = 111 \implies b + e = 74 Col 3: 7 + c + f = 111 \implies 7 + 61 + f = 111 \implies f = 43 Step 3. Diagonal sums: Diagonal 1 (top-left to bottom-right): a + 37 + f = 111 \implies a + 37 + 43 = 111 \implies a = 31 Diagonal 2 (top-right to bottom-left): 7 + 37 + d = 111 \implies d = 67 Step 4. From a + b = 104 and a = 31, find b: 31 + b = 104 \implies b = 73 Step 5. From a + d = 98 and a=31, d=67 (consistent) Step 6. From b + e = 74 and b=73, e=1 Step 7. From row 3 sum: d + e + f = 67 + 1 + 43 = 111 (correct) Filled magic square: $$\begin{matrix} 31 & 73 & 7 \\ 13 & 37 & 61 \\ 67 & 1 & 43 \end{matrix}$$ Answer verified: all rows, columns, diagonals sum to 111. --- "slug":"sequence-magic-square", "subject":"logic and algebra", "desmos":{"latex":"", "features": {"intercepts": false, "extrema": false}}, "q_count":6