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Puzzles

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Key numbers

Key answers

Ways for the first row of a 3 × 3 Sudoku6 (class 6, page 114)
Solutions of a 3 × 3 Sudoku12 (class 6, page 114-115)
1-6 magic triangle, exampleside sum 12 (class 6, page 116)
1-6 magic triangle, exerciseside sum 9 (class 6, page 130)
Side sums of the 1-9 magic triangle17, 19, 20, 21, 23 (class 6, page 130)
Magic triangle of odd numbers 1-17side sum 30 (class 6, page 130)
1-7 with 7 in the centreeach line 14 (class 6, page 130)
1-12 stareach line 26 (class 6, page 130)
4 × 4 magic square from tetrominoesevery row, column and diagonal 34 (class 7, page 137)
The five tetrominoes used twice5 × 8 and 4 × 10 rectangles (class 7, page 120)

Easy to confuse

நிரை / நிரல்horizontal line (row) / vertical line (column)
4 × 4 Sudoku / 3 × 3 Sudokuthe 2 × 2 box rule is extra in the 4 × 4 only; the 3 × 3 has no such rule
3 × 3: first-row ways / solutions6 / 12
1-6 magic triangle: 12 / 9both are right (page 116, page 130); with no sum stated there is no single answer
1-9 magic triangle: "5" / 18"5" counts the kinds of side sum; the distinct arrangements number 18
One arrangement of a magic square / its sumthere are many arrangements; only the sum 34 (26 for the star) is certain
Load puzzle: stage by stage / gates × starting sacksthe toll falls as the sacks fall; one product is wrong
"Immediately left" / "somewhere left"only "immediately" gives a single answer; the key follows that sense
The meaning of "Sudoku" / a questionthe book's note; no primary source; "who invented it" is not a fact for questions
The full note

Completing a systematic list

To fill the missing numbers of a list, first see how the filled part is arranged. Then use the same arrangement for the rest. This is the base of every puzzle. Example: if some names in a school attendance register are in alphabetical order, the missing name must fit the same order.

Sudoku

Sudoku is a number game; some cells already have numbers. The horizontal cells are a row (நிரை); the vertical cells are a column (நிரல்). In a 9 × 9 Sudoku the numbers 1 to 9 may not appear more than once in a row or a column; a 3 × 3 uses only 1-3 and a 4 × 4 only 1-4. As the book prints it, "Sudoku" is a Japanese word: "Su" means number and "Doku" single. This note and the name of its inventor have no primary source, so they are not facts for questions.

Ways in a 3 × 3 Sudoku

The first row of a 3 × 3 Sudoku can be filled in 6 ways: (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1). If the first row is fixed as 1 2 3, the second row can only be 2 3 1 or 3 1 2. Once the first two rows are fixed, the third row has only one way. So there are 6 × 2 = 12 solutions; the book shows 12 solutions, (அ) to (ஔ). Do not confuse "6 first rows" with "12 solutions".

The 4 × 4 Sudoku

A 4 × 4 Sudoku has one more rule: in each of its four 2 × 2 boxes no number from 1 to 4 may repeat. The 3 × 3 Sudoku has no such rule. The book's solution, row by row: 1 4 3 2 / 3 2 1 4 / 4 1 2 3 / 2 3 4 1. Every row, column and 2 × 2 box has 1-4 exactly once. Example: the top-left 2 × 2 box holds 1, 4, 3, 2; the first column holds 1, 3, 4, 2.

The magic triangle

In a magic triangle each number is used only once and the sum of every side of the triangle must be the same. For the six numbers 1 to 6 there are six places: the three corners and the midpoints of the three sides. For a side sum of 12, step 1: put the large numbers 4, 5, 6 at the corners; step 2: put the small numbers 1, 2, 3 in the middles of the sides. Placing the middle numbers wrongly gives unequal side sums such as 10, 13, 13 or 11, 14, 11.

Example: A magic triangle with side sum 12

Make a magic triangle with the numbers 1 to 6 in which every side adds up to 12.

  1. Corners: 4, 5, 6.
  2. 3 between 4 and 5; 1 between 5 and 6; 2 between 4 and 6.
  3. Check: 4 + 3 + 5 = 12, 5 + 1 + 6 = 12, 4 + 2 + 6 = 12.

Answer: all three sides are 12. With 1, 3, 2 at the corners and 5, 4, 6 in the middles you also get a magic triangle with side sum 9 (1 + 5 + 3 = 3 + 4 + 2 = 2 + 6 + 1 = 9).

Other magic triangle answers

A magic triangle with 1 to 6 has no single side sum: the book's example has 12 and the answers page 9. So a question that does not state the sum has no single answer. A magic triangle can be made with 1 to 9 (four to a side); the side sums are 17, 19, 20, 21, 23. The book's "5" is the number of these sums; the distinct arrangements number 18. With the odd numbers from 1 to 17 the side sum is 30: the sides are 1, 17, 7, 5; 1, 11, 15, 3; 5, 13, 9, 3. With 1 to 7 and 7 in the centre, every straight line adds up to 14.

The star and the magic square

The book has a puzzle that puts the numbers 1 to 12, once each, in the 12 circles of a star so that each of the 6 lines adds up to 26; the answer says "there are many ways". The 4 × 4 magic square built from tetrominoes, row by row, is 16 3 2 13 / 5 10 11 8 / 9 6 7 12 / 4 15 14 1. Every row, column and diagonal adds up to 34. These too have "more possible ways". So do not treat one arrangement as the only right one; only the sum (26, 34) is certain.

Rectangles from tetrominoes

The five tetrominoes together make 20 squares. So you may try rectangles such as 1 × 20, 2 × 10 and 5 × 4. But using each one only once, no rectangle can be filled completely. The reason: if the squares are coloured alternately like a chessboard, four of the tetrominoes have equal numbers of coloured and uncoloured squares; only the T-shaped one does not. Using all five twice fills 5 × 8 and 4 × 10 rectangles completely. Example: the same thinking helps when laying square tiles on a floor or packing many things into a small box.

The load puzzle

In some puzzles a toll of one item is paid at each gate for every sack carried. Whenever what is left fits into fewer sacks, it can be repacked and the empty sacks dropped. The toll at the later gates then falls. So work out each stage in which the number of sacks changes separately. Do not work it out as one product, "gates × starting sacks".

Example: Two sacks, 12 gates

2 sacks hold 10 items each. At each gate a toll of 1 item is paid for every sack carried. How many are left after 12 gates?

  1. Stage one: 2 sacks, 2 per gate. After 5 gates, 20 − 10 = 10 are left; they fit in one sack.
  2. Stage two: 1 sack, 1 per gate. The remaining 7 gates take 7.
  3. 10 − 7 = 3.

Answer: 3 items. Paying 2 at every one of the 12 gates comes to more than 20 and gives the wrong answer "nothing is left".

The seating puzzle

In a seating question, place the fixed conditions (centre, ends) first, then fill the rest. In a question where a single answer comes only when "to the left" is read as "immediately to the left" (the next seat), the key follows that sense. So number the seats 1, 2, 3 from left to right and write it out. Check whether reading "somewhere to the left" gives two arrangements.

Example: Five people in a row

Arun, Bala, Sudha, Devi and Ezhil sit in seats 1 to 5 from left to right. Sudha is in the centre; Arun and Ezhil are at the two ends; Devi is immediately to the left of Ezhil. Who is immediately to the right of Arun?

  1. Sudha is 3. Arun and Ezhil are 1 or 5.
  2. If Ezhil were at 1 there would be no seat to the left; so Ezhil is 5, Devi 4 and Arun 1.
  3. Bala takes the remaining seat 2.

Answer: Bala. Without "immediately", "somewhere to the left" lets Devi sit at 2 or 4, and there is no single answer.

Source: கணக்கு, ஆறாம் வகுப்பு, முதல் பருவம், தொகுதி 2, தமிழ் வழி (Mathematics, Standard VI, Term I, Volume 2, Tamil medium; Samacheer Kalvi) (6ஆம் வகுப்புக் கணக்கு, பருவம் 1, இயல் 6 (6.3 முறையான பட்டியலை முழுமையாக்குதல்), பக்கம் 113-118, விடைகள் பக்கம் 130; 7ஆம் வகுப்புக் கணக்கு, பருவம் 1, இயல் 6 (6.2 நாற்சதுர இணை), பக்கம் 119-120, 124, விடைகள் பக்கம் 137; TNPSC வினாத்தாள்கள் CCSE4GT/2024 பக்கம் 59, CCSEIVGT/2025 பக்கம் 122 (வினாத்தாள் விதிகள் மட்டும்))

PYQ (2024-2025 papers)

Free questions

  1. Question 1

    Of the 12 solutions of a 3 × 3 Sudoku, how many have 1 in the top-left cell?

    1. A. 2
    2. B. 6
    3. C. 3
    4. D. 4
    Answer and explanation

    Correct answer: D. 4

    Two first rows begin with 1, (1, 2, 3) and (1, 3, 2); each has 2 second rows and 1 third row; so 2 × 2 = 4 (a third of 12).

    Source: கணக்கு, ஆறாம் வகுப்பு, முதல் பருவம், தொகுதி 2, தமிழ் வழி (Mathematics, Standard VI, Term I, Volume 2, Tamil medium; Samacheer Kalvi) (இயல் 6 தகவல் செயலாக்கம் (6.3.1 சுடோகு), pp. 114-115 (PDF pp. 118-119))

  2. Question 2

    A list of the numbers written with the digits 1, 2, 3, each once, runs systematically as 123, 132, 213, 231, 312, ... Which number completes the list?

    1. A. 321
    2. B. 313
    3. C. 213
    4. D. 331
    Answer and explanation

    Correct answer: A. 321

    The list is in increasing order: starting with 1, 123 and 132; with 2, 213 and 231; with 3, 312 and 321; so the missing number is 321.

    Source: கணக்கு, ஆறாம் வகுப்பு, முதல் பருவம், தொகுதி 2, தமிழ் வழி (Mathematics, Standard VI, Term I, Volume 2, Tamil medium; Samacheer Kalvi) (இயல் 6 தகவல் செயலாக்கம் (6.3 முறையான பட்டியலை முழுமையாக்குதல்), p. 113 (PDF p. 117))

  3. Question 3

    A list of the pairs that can be made from the four people A, B, C, D runs systematically as AB, AC, AD, BC, BD, ... Which pair completes the list?

    1. A. BA
    2. B. CB
    3. C. CD
    4. D. DA
    Answer and explanation

    Correct answer: C. CD

    Each letter is paired only with the letters after it: A with B, C, D; B with C, D; C with D; so the last pair is CD (6 pairs in all).

    Source: கணக்கு, ஆறாம் வகுப்பு, முதல் பருவம், தொகுதி 2, தமிழ் வழி (Mathematics, Standard VI, Term I, Volume 2, Tamil medium; Samacheer Kalvi) (இயல் 6 தகவல் செயலாக்கம் (6.3 முறையான பட்டியலை முழுமையாக்குதல்), p. 113 (PDF p. 117))

  4. Question 4

    Match the following on Sudoku. (a) Row (நிரை) - 1. The numbers 1 to 4 (b) Column (நிரல்) - 2. Vertical cells (c) 3 × 3 Sudoku - 3. The numbers 1 to 3 (d) 4 × 4 Sudoku - 4. Horizontal cells

    1. A. (a) 2, (b) 4, (c) 3, (d) 1
    2. B. (a) 4, (b) 2, (c) 3, (d) 1
    3. C. (a) 4, (b) 2, (c) 1, (d) 3
    4. D. (a) 4, (b) 3, (c) 2, (d) 1
    Answer and explanation

    Correct answer: B. (a) 4, (b) 2, (c) 3, (d) 1

    Horizontal cells form a row and vertical cells a column; a 3 × 3 Sudoku uses only 1-3 and a 4 × 4 Sudoku only 1-4; so (a) 4, (b) 2, (c) 3, (d) 1.

    Source: கணக்கு, ஆறாம் வகுப்பு, முதல் பருவம், தொகுதி 2, தமிழ் வழி (Mathematics, Standard VI, Term I, Volume 2, Tamil medium; Samacheer Kalvi) (இயல் 6 தகவல் செயலாக்கம் (6.3.1 சுடோகு), p. 113 (PDF p. 117))

The unit's 20 free questions

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Frequently asked questions

Make a magic triangle with the numbers 1 to 6 in which every side adds up to 12.

all three sides are 12. With 1, 3, 2 at the corners and 5, 4, 6 in the middles you also get a magic triangle with side sum 9 (1 + 5 + 3 = 3 + 4 + 2 = 2 + 6 + 1 = 9).

2 sacks hold 10 items each. At each gate a toll of 1 item is paid for every sack carried. How many are left after 12 gates?

3 items. Paying 2 at every one of the 12 gates comes to more than 20 and gives the wrong answer "nothing is left".

Arun, Bala, Sudha, Devi and Ezhil sit in seats 1 to 5 from left to right. Sudha is in the centre; Arun and Ezhil are at the two ends; Devi is immediately to the left of Ezhil. Who is immediately to the right of Arun?

Bala. Without "immediately", "somewhere to the left" lets Devi sit at 2 or 4, and there is no single answer.

Full guide: TNPSC Group 4 Part C Aptitude: 25 questions, 2 units and how to practise them

Sources

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