Srinivasa Ramanujan
Every year on 22 December schools celebrate National Mathematics Day. It is the birthday of a boy from a small town who loved numbers so much that he became world famous.
Srinivasa Ramanujan (श्रीनिवास रामानुजन) was a great Indian mathematician who worked on number theory, analysis and infinite series.
Rule 1: His life — Ramanujan was born in Erode (then in the Madras province) on 22-12-1887 and died on 26-04-1920. He was about 32 years old.
In the figure : In the figure the Ramanujan timeline.
- Easy : Easy: Born in 1887, died in 1920.
- Medium : Medium: 1920 − 1887 = 33, but his birthday (December) had not come in April 1920, so his age was 32.
- Exam-type : Exam: He was born in Erode, in the Madras province of that time.
Rule 2: His work and National Mathematics Day — He became famous for proving about 4000 theorems. His birthday, 22 December, is celebrated every year as National Mathematics Day. From 1887 to 2026 is 139 years.
- Easy : Easy: 2026 − 1887 = 139 years.
- Medium : Medium: His 100th birth anniversary was in 1887 + 100 = 1987.
- Exam-type : Exam: His 150th birth anniversary will come in 1887 + 150 = 2037.
Born 22-12-1887, Erode. Died 26-04-1920. National Mathematics Day: 22 December.
Finding age as 1920 − 1887 = 33 without checking the birthday month.
Memory clue: "बावीस बारा — गणित दिन" (22 December).
Spot the difference
| Fact | Value |
|---|---|
| Born | 22-12-1887 |
| Died | 26-04-1920 |
| Theorems | about 4000 |
Practice questions
- 1When was Ramanujan born?
- 2Where was he born?
- 3Which day is National Mathematics Day?
- 4At what age did he die?
- 5Name two fields he worked in.
- 6How many years from his birth in 1887 to 2026?
- 7About how many theorems did he prove?
- 8In which year was his 100th birth anniversary?
Show answers (try them yourself first)
- 122-12-1887.
- 2Erode.
- 322 December.
- 4About 32 years.
- 5Number theory and infinite series (or analysis).
- 62026 − 1887 = 139.
- 7About 4000.
- 81987.
Ramanujan square
Ramanujan wrote his own date of birth as the first row of a number square, and the square has a wonderful property. Every line adds to the same number.
In a magic square (जादुई चौरस) every row, every column and both diagonals add up to the same number, called the magic sum.
Rule 1: The Ramanujan square — Its rows are 22, 12, 18, 87 / 88, 17, 9, 25 / 10, 24, 89, 16 / 19, 86, 23, 11. The first row is the date 22 12 18 87. Every line adds to 139.
In the figure : In the figure the Ramanujan magic square.
- Easy : Easy: Row 2: 88 + 17 + 9 + 25 = 139.
- Medium : Medium: Column 2: 12 + 17 + 24 + 86 = 139.
- Exam-type : Exam: Diagonal: 87 + 9 + 24 + 19 = 139. The other diagonal: 22 + 17 + 89 + 11 = 139.
Rule 2: Finding missing numbers — A missing number = magic sum − the other three. In row 2, if 17 is missing: 139 − (88 + 9 + 25) = 17. All 16 numbers add up to 4 × 139 = 556.
- Easy : Easy: Row 3 with 89 missing: 139 − (10 + 24 + 16) = 89.
- Medium : Medium: Column 3 with 9 missing: 139 − (18 + 89 + 23) = 9.
- Exam-type : Exam: The 16 numbers form 4 rows of 139, so their total is 4 × 139 = 556.
Ramanujan square
In the app you can move this figure with your finger.
Open in the appMagic sum 139. Rows, columns, diagonals. Missing number = 139 − the other three.
Forgetting the diagonals. They also add up to 139.
Memory clue: "सर्व रेषा १३९" (every line 139).
Spot the difference
| Line | Sum |
|---|---|
| Each row | 139 |
| Each column | 139 |
| Each diagonal | 139 |
Practice questions
- 1What is the sum of the first row 22, 12, 18, 87?
- 2What is the magic sum of the Ramanujan square?
- 3How many rows does a 4 × 4 square have?
- 4Find ?: 19, 86, ?, 11.
- 5Find ?: column 87, 25, 16, ?.
- 6What is the total of all 16 numbers?
- 7Verify: 22 + 17 + 89 + 11.
- 8Why is the first row of the square special?
Show answers (try them yourself first)
- 1139.
- 2139.
- 34.
- 4139 − 116 = 23.
- 5139 − 128 = 11.
- 64 × 139 = 556.
- 7139, a diagonal.
- 8It is Ramanujan's date of birth, 22 12 18 87.
Tower of Hanoi (Brahma Tower)
In an old legend, priests moved 64 golden discs from one pole to another to finish the world's work. The puzzle based on it is the Tower of Hanoi, also called the Brahma Tower.
The Tower of Hanoi (हनोईचा मनोरा) has three sticks and discs of different sizes stacked largest at the bottom. The aim is to move the whole stack to another stick.
Rule 1: Rules — (1) Move one disc at a time. (2) Never put a larger disc on a smaller disc. (3) At the end all discs must be on stick 3 in the original order.
In the figure : In the figure the Tower of Hanoi.
- Easy : Easy: With 1 disc it takes 1 move.
- Medium : Medium: With 2 discs it takes 3 moves: small to stick 2, large to stick 3, small to stick 3.
- Exam-type : Exam: With 3 discs the minimum is 7 moves.
Rule 2: The move-count pattern — Minimum moves: 1 disc → 1, 2 → 3, 3 → 7, 4 → 15, 5 → 31, 6 → 63, 7 → 127. Each extra disc doubles the moves and adds 1, so for n discs t = 2ⁿ − 1.
- Easy : Easy: 4 discs need 2 × 7 + 1 = 15 moves.
- Medium : Medium: 5 discs need 2 × 15 + 1 = 31 moves.
- Exam-type : Exam: 8 discs need 2⁸ − 1 = 256 − 1 = 255 moves.
Tower of Hanoi (Brahma Tower)
In the app you can move this figure with your finger.
Open in the appOne disc at a time. Large never on small. t = 2ⁿ − 1.
Placing a larger disc on a smaller disc. That breaks the rule.
Memory clue: "चकत्या n, हालचाली दोनाचा n-वा घात वजा एक".
Spot the difference
| Discs | Minimum moves |
|---|---|
| 1 | 1 |
| 2 | 3 |
| 3 | 7 |
| 4 | 15 |
Practice questions
- 1How many moves for 1 disc?
- 2How many moves for 2 discs?
- 3How many moves for 3 discs?
- 4How many moves for 4 discs?
- 5How many moves for 6 discs?
- 6How many moves for 7 discs?
- 7How many moves for 8 discs?
- 8If one move takes 1 second, how long will 5 discs take?
Show answers (try them yourself first)
- 11.
- 23.
- 37.
- 415.
- 563.
- 6127.
- 72⁸ − 1 = 255.
- 831 moves = 31 seconds.
Finding the lighter object with a pan balance
A shopkeeper has 9 packets that look the same, but one packet is a little lighter. Using a pan balance only, he can find the lighter packet in just 2 weighings.
On a pan balance (पारड्याचा तराजू) the pan that goes up holds the lighter object. If both pans stay level, the objects weigh the same.
Rule 1: Finding the lighter one among 3 — Weigh 1 against 1. If one pan goes up, that object is lighter. If they are level, the third object (kept aside) is the lighter one. So 1 weighing finds the lighter among 3 objects.
In the figure : In the figure the pan balance method.
- Easy : Easy: Pan A goes up, so the object in pan A is lighter.
- Medium : Medium: Both pans stay level, so the two objects weigh the same.
- Exam-type : Exam: Among 3 identical-looking objects, 1 weighing is enough to find the lighter one.
Rule 2: 9 and 27 objects — Among 9 objects weigh 3 against 3. If level, the lighter one is among the 3 kept aside; otherwise it is in the pan that went up. Then weigh 1 against 1 from that group of 3. Only 2 weighings are needed. Pattern: 1 weighing handles 3 objects, 2 handle 9, 3 handle 27 (3 × 3 × 3).
- Easy : Easy: 9 objects need 2 weighings.
- Medium : Medium: 27 objects need 3 weighings.
- Exam-type : Exam: 81 objects need 4 weighings because 3 × 3 × 3 × 3 = 81.
Up = lighter. Level = equal. Divide in three equal groups.
Dividing into two groups. Three groups are much faster.
Memory clue: "तीन गट, तीन पट" (three groups, three times as many).
Spot the difference
| Weighings | Objects handled |
|---|---|
| 1 | 3 |
| 2 | 9 |
| 3 | 27 |
| 4 | 81 |
Practice questions
- 1Which pan holds the lighter object?
- 2What does it mean if both pans are level?
- 3How many weighings find the lighter among 3 objects?
- 4How many weighings for 9 objects?
- 5How many weighings for 27 objects?
- 6How many weighings for 81 objects?
- 7Describe the steps for 9 objects.
- 8Can 10 objects be solved in 2 weighings? Why?
Show answers (try them yourself first)
- 1The pan that goes up.
- 2The objects weigh the same.
- 31.
- 42.
- 53.
- 64.
- 7Weigh 3 against 3; the lighter group is the up pan (or the 3 kept aside if level); then weigh 1 against 1 from that group.
- 8Not always; 2 weighings handle at most 9 objects.
Sum of areas of squares
Two square rugs of different sizes can be cut and joined into one bigger square rug. No cloth is wasted, so the areas add up.
When two squares are cut and rearranged into one square, the new square's area is the sum of the two areas. The perimeters do not add.
Rule 1: Joining two squares — Take a square of side 8 (area 64) and a square of side 4 (area 16). Join them and cut along the marked lines. The pieces make one new square of area 64 + 16 = 80. The areas add up, but 8 + 4 is not the new side.
In the figure : In the figure joining two squares.
- Easy : Easy: Squares of side 2 and 1 give new area 4 + 1 = 5.
- Medium : Medium: Squares of side 6 and 8: 36 + 64 = 100, so the new square has side 10.
- Exam-type : Exam: Perimeters 32 and 16 add up to 48, but the new square's perimeter is smaller; perimeters do not add.
Rule 2: The Baudhayana-Pythagoras idea — With sides 3 and 4 the new square has area 9 + 16 = 25, so its side is 5. In the same way 5 and 12 give 25 + 144 = 169, side 13.
- Easy : Easy: 3² + 4² = 9 + 16 = 25 = 5².
- Medium : Medium: 6² + 8² = 36 + 64 = 100 = 10².
- Exam-type : Exam: 5² + 12² = 25 + 144 = 169 = 13².
Areas add. Perimeters do not add. 3, 4, 5 is the first example.
Adding the sides: 3 + 4 = 7 as the new side. The new side is 5.
Memory clue: "क्षेत्रफळ + क्षेत्रफळ = नवे क्षेत्रफळ".
Spot the difference
| Sides | New square |
|---|---|
| 3 and 4 | Area 25, side 5 |
| 6 and 8 | Area 100, side 10 |
| 5 and 12 | Area 169, side 13 |
Practice questions
- 1Squares of side 2 and 1 are joined. Find the new area.
- 2Squares of side 3 and 4. Find the new area.
- 3What is the side of a square of area 25?
- 4Squares of side 6 and 8. Find the new area and side.
- 5Squares of side 5 and 12. Find the new side.
- 6Squares of side 8 and 4. Find the new area.
- 7Do the perimeters of the two squares add up to the perimeter of the new square?
- 8Why is 3 + 4 = 7 not the side of the new square?
Show answers (try them yourself first)
- 14 + 1 = 5.
- 29 + 16 = 25.
- 35.
- 4100 and 10.
- 525 + 144 = 169, side 13.
- 664 + 16 = 80.
- 7No. Only the areas add up.
- 8The areas add (9 + 16 = 25), not the sides; the side is 5.