The smart cards trick
A magician shows you five cards and asks, "On which cards is your secret number?" In a second he tells your number. There is no magic, only clever addition.
Five smart cards hold the numbers 1 to 31. To guess a number, add the first numbers of the cards that contain it. The first numbers are 1, 2, 4, 8 and 16.
Rule 1: The trick — Ask which cards contain the secret number and add the first numbers of those cards. If the number is on the cards starting with 2, 4 and 16, the number is 2 + 4 + 16 = 22.
In the figure : In the figure the smart cards: 13 = 1 + 4 + 8.
- Easy : Easy: Cards starting 1 and 2: the number is 1 + 2 = 3.
- Medium : Medium: Cards starting 1, 8 and 16: 1 + 8 + 16 = 25.
- Exam-type : Exam: Cards starting 2, 4, 8 and 16: 2 + 4 + 8 + 16 = 30.
Rule 2: Special cases — If the number is on all five cards, it is 1 + 2 + 4 + 8 + 16 = 31. If it is on only one card, the number is that card's first number.
- Easy : Easy: Only on the card starting 8: the number is 8.
- Medium : Medium: 10 is on the cards starting 2 and 8, because 2 + 8 = 10.
- Exam-type : Exam: The number 31 is on every card because 1 + 2 + 4 + 8 + 16 = 31.
Add the first numbers of the chosen cards. First numbers: 1, 2, 4, 8, 16. Maximum 31.
Adding all numbers on the cards, not only the first ones. Add only the first numbers.
Memory clue: "पहिले आकडे जोडा, गुप्त संख्या मिळवा".
Spot the difference
| Cards starting | Number |
|---|---|
| 1, 2 | 3 |
| 2, 4, 16 | 22 |
| 1, 2, 4, 8, 16 | 31 |
Practice questions
- 1Cards starting 1 and 2 contain the number. Find it.
- 2Cards starting 4 and 8. Find the number.
- 3Cards starting 1, 4 and 16. Find the number.
- 4Cards starting 2, 8 and 16. Find the number.
- 5The number is on all five cards. What is it?
- 6On which cards is 11?
- 7On which cards is 20?
- 8Ravi says his number is on no card. Which number could it be?
Show answers (try them yourself first)
- 11 + 2 = 3.
- 212.
- 321.
- 426.
- 531.
- 611 = 8 + 2 + 1, so the cards starting 1, 2 and 8.
- 720 = 16 + 4, so the cards starting 4 and 16.
- 80, since the sum of no first numbers is 0 (or a number above 31).
Angles of the star and polygons
The tips of a starfish and the corners of a nut have angles that follow neat rules. If we know that a triangle's angles add up to 180°, we can find the angles of every polygon.
Angles on a straight line add up to 180°, and the angles of a triangle add up to 180°. A polygon with n sides has angle sum (n − 2) × 180°.
Rule 1: Angle sum of a polygon — Cut a polygon into triangles from one corner. A quadrilateral gives 2 triangles, so 2 × 180° = 360°. A regular hexagon gives 4 triangles, so 4 × 180° = 720°, and each angle is 720° ÷ 6 = 120°.
- Easy : Easy: A quadrilateral has angle sum 360°.
- Medium : Medium: A regular octagon: (8 − 2) × 180° = 1080°, so each angle is 1080° ÷ 8 = 135°.
- Exam-type : Exam: A polygon has angle sum 1440°. Then (n − 2) × 180° = 1440°, n − 2 = 8, so n = 10 sides.
Rule 2: The star from a pentagon — A regular pentagon has 3 triangles, so the angle sum is 3 × 180° = 540° and each angle is 540° ÷ 5 = 108°. Extend its sides to make a star. The outside angle is 180° − 108° = 72°, so each tip is 180° − 72° − 72° = 36°.
In the figure : In the figure a star with 36° tips and a 108° pentagon.
- Easy : Easy: A regular hexagon has outside angle 180° − 120° = 60°.
- Medium : Medium: Extending the sides of a regular hexagon gives tips of 180° − 60° − 60° = 60°.
- Exam-type : Exam: The five tips of the pentagon star add up to 5 × 36° = 180°.
Triangle 180°. Polygon (n − 2) × 180°. Regular polygon: divide by n for each angle.
Using n × 180° for the angle sum. Use (n − 2) × 180°.
Memory clue: "बाजू वजा दोन, गुणिले एकशे ऐंशी".
Spot the difference
| Polygon | Angle sum |
|---|---|
| Triangle | 180° |
| Quadrilateral | 360° |
| Pentagon | 540° |
| Hexagon | 720° |
Practice questions
- 1What is the angle sum of a triangle?
- 2What is the angle sum of a quadrilateral?
- 3What is the angle sum of a pentagon?
- 4Find each angle of a regular hexagon.
- 5Find each angle of a regular octagon.
- 6Find the angle sum of a heptagon (7 sides).
- 7A polygon has angle sum 1260°. How many sides does it have?
- 8In the pentagon star, why is each tip 36°?
Show answers (try them yourself first)
- 1180°.
- 2360°.
- 3540°.
- 4720° ÷ 6 = 120°.
- 51080° ÷ 8 = 135°.
- 6(7 − 2) × 180° = 900°.
- 7n − 2 = 7, so n = 9.
- 8The outside angle is 180° − 108° = 72°, and the tip is 180° − 72° − 72° = 36°.
Pentominoes and tiling
Building blocks that join edge to edge can make many shapes. A pentomino uses exactly five equal squares. These shapes make wonderful puzzles.
A pentomino (पेंटोमिनो) joins 5 equal squares edge to edge. There are exactly 12 of them. Tiling (फरशीकाम) means covering a region with no overlap and no gap.
Rule 1: The twelve pentominoes — They are named after letters: F, I, L, N, P, T, U, V, W, X, Y, Z. Each has area 5, so all 12 together have area 5 × 12 = 60.
In the figure : In the figure the 12 pentominoes.
- Easy : Easy: The I pentomino is 5 squares in a row, a 1 × 5 rectangle.
- Medium : Medium: Two pentominoes have area 10; they can make a 2 × 5 rectangle.
- Exam-type : Exam: The rectangles 6 × 10, 5 × 12, 4 × 15 and 3 × 20 all have area 60, the area of all 12 pentominoes.
Rule 2: Tiling and the area rule — Each pentomino covers 5 squares, so the area covered must be a multiple of 5. A 5 × 4 grid (area 20) may be covered by 4 pentominoes. A 7 × 8 grid (area 56) can never be fully covered.
- Easy : Easy: A 5 × 4 grid has area 20 = 4 × 5, so four pentominoes might fit.
- Medium : Medium: A 6 × 5 grid has area 30 = 6 × 5, so six pentominoes might fit.
- Exam-type : Exam: A 7 × 8 grid has area 56, which is not a multiple of 5, so it cannot be tiled by pentominoes.
5 squares each. 12 shapes. Area covered = multiple of 5.
Saying a grid can be tiled just because the area is a multiple of 5. It is necessary but not always enough.
Memory clue: "पाच चौरस, बारा आकार, क्षेत्रफळ पाचाचे पट".
Spot the difference
| Item | Value |
|---|---|
| Squares in a pentomino | 5 |
| Number of pentominoes | 12 |
| Total area of all | 60 |
Practice questions
- 1How many squares are in a pentomino?
- 2How many different pentominoes are there?
- 3What is the area of one pentomino?
- 4What is the total area of all 12 pentominoes?
- 5Name two rectangles that have area 60.
- 6Can two pentominoes cover a 2 × 5 rectangle?
- 7Can a 7 × 7 grid be fully covered by pentominoes?
- 8Why must the covered area be a multiple of 5?
Show answers (try them yourself first)
- 15.
- 212.
- 35 square units.
- 412 × 5 = 60.
- 56 × 10 and 5 × 12.
- 6Yes, area 10 = 2 × 5; for example two L-shapes fit.
- 7No, area 49 is not a multiple of 5.
- 8Every pentomino covers exactly 5 squares.
Addition and subtraction grids
In a puzzle grid, numbers must add up across rows. In a sum pyramid, every brick is the sum of the two bricks below it. These puzzles train quick, careful addition.
In an addition grid the row total is the sum of its numbers. In a subtraction cell the answer is the difference. In a sum pyramid (बेरीज पिरॅमिड) each cell is the sum of the two cells below it.
Rule 1: Grids — If a row has 7, 2 and a missing number with total 15, the missing number is 15 − 7 − 2 = 6. In a subtraction cell, 12 − 5 = 7.
- Easy : Easy: Row 5, ?, 4 with total 12: ? = 12 − 5 − 4 = 3.
- Medium : Medium: A cell 9 − 4 = 5.
- Exam-type : Exam: Row total 20 with 8, ?, 5: ? = 20 − 8 − 5 = 7.
Rule 2: Sum pyramid — Bottom row 6, 2, 5 gives the middle row 6 + 2 = 8 and 2 + 5 = 7, and the top is 8 + 7 = 15. The same works with fractions: bottom 14 and 18 give 28 + 18 = 38.
In the figure : In the figure a sum pyramid.
- Easy : Easy: Bottom 3 and 4 give top 3 + 4 = 7.
- Medium : Medium: Bottom 2, 6, 1 give 8 and 7, so the top is 15.
- Exam-type : Exam: Top is 21 and the bottom is 5, ?, 4. Middle cells are 5 + x and x + 4; top = 9 + 2x = 21, so x = 6.
Grid: total minus the others. Pyramid: add the two below. Missing bottom: work with a letter.
Adding the wrong pair of cells. Each top cell uses the two cells directly below it.
Memory clue: "दोन खालची = एक वरची".
Spot the difference
| Puzzle | Rule |
|---|---|
| Addition grid | Row total = sum |
| Subtraction cell | Answer = difference |
| Sum pyramid | Cell = two below |
Practice questions
- 1Find ?: 4 + ? = 9.
- 2A row 6, 3, ? has total 14. Find ?.
- 3Bottom 2 and 3 of a pyramid. Find the top.
- 4Bottom 1, 4, 2. Find the top.
- 5Bottom 5, 1, 7. Find the top.
- 6Add the pyramid cells 12 and 14.
- 7Top 17, bottom 3, ?, 6. Find ?.
- 8Three equal numbers in a row have total 24. Find each.
Show answers (try them yourself first)
- 15.
- 214 − 9 = 5.
- 35.
- 4Middle 5 and 6, top 11.
- 5Middle 6 and 8, top 14.
- 624 + 14 = 34.
- 73 + 2x + 6 = 17, so x = 4. Check: 7, 10, top 17 ✓.
- 824 ÷ 3 = 8.
Why the trick works
With weights of 1, 2, 4, 8 and 16 kg you can weigh any whole number of kilograms up to 31 using each weight at most once. The smart cards work for the same reason.
The first numbers 1, 2, 4, 8, 16 each double the one before. Every number up to 31 is a sum of some of them, each used at most once.
Rule 1: Why it works — Write a number as a sum of the doubling numbers: 19 = 16 + 2 + 1 and 27 = 16 + 8 + 2 + 1. So 19 sits exactly on the cards starting 1, 2 and 16.
In the figure : In the figure the doubling numbers 1, 2, 4, 8, 16.
- Easy : Easy: 5 = 4 + 1, so 5 is on the cards 1 and 4.
- Medium : Medium: 14 = 8 + 4 + 2, so 14 is on the cards 2, 4 and 8.
- Exam-type : Exam: 27 = 16 + 8 + 2 + 1 sits on four cards: 1, 2, 8, 16.
Rule 2: More cards, bigger numbers — With 3 cards the largest number is 1 + 2 + 4 = 7, with 4 cards it is 15, with 5 cards it is 31, and with 6 cards (adding 32) it is 63. Each time the largest is one less than a power of 2.
- Easy : Easy: With 3 cards: 1 + 2 + 4 = 7.
- Medium : Medium: With 4 cards: 1 + 2 + 4 + 8 = 15.
- Exam-type : Exam: With 7 cards (adding 64) the largest is 63 + 64 = 127.
Doubling numbers: 1, 2, 4, 8, 16, 32. n cards → largest number is one less than the next doubling number.
Using a doubling number twice. Each is used at most once.
Memory clue: "१, २, ४, ८, १६ — प्रत्येक पुढची दुप्पट".
Spot the difference
| Cards | Largest number |
|---|---|
| 3 | 7 |
| 4 | 15 |
| 5 | 31 |
| 6 | 63 |
Practice questions
- 1Write 6 as a sum of doubling numbers.
- 2Write 9 as a sum of doubling numbers.
- 3Write 21 as a sum of doubling numbers.
- 4Write 30 as a sum of doubling numbers.
- 5What is the largest number on 3 cards?
- 6What is the largest number on 6 cards?
- 7How many cards are needed for numbers up to 100?
- 8Why can't the number 32 be on the five cards?
Show answers (try them yourself first)
- 14 + 2.
- 28 + 1.
- 316 + 4 + 1.
- 416 + 8 + 4 + 2.
- 57.
- 663.
- 77 cards (largest 127), because 6 cards stop at 63.
- 8The five cards add up to at most 31, so 32 needs a sixth card starting with 32.