House of Hundreds - II

Given number: 832

We break 832 into hundreds, tens, and ones:
$$832 = 800 + 30 + 2$$

- Hundreds: 8 big tiles (each representing 100)
- Tens: 3 medium tiles (each representing 10)
- Ones: 2 small tiles (each representing 1)

Draw 8 hundred-tiles, 3 ten-tiles, and 2 one-tiles in your notebook.

Given number: 947

We break 947 into hundreds, tens, and ones:
$$947 = 900 + 40 + 7$$

- Hundreds: 9 big tiles
- Tens: 4 medium tiles
- Ones: 7 small tiles

Draw 9 hundred-tiles, 4 ten-tiles, and 7 one-tiles in your notebook.

Given number: 726

We break 726 into hundreds, tens, and ones:
$$726 = 700 + 20 + 6$$

- Hundreds: 7 big tiles
- Tens: 2 medium tiles
- Ones: 6 small tiles

Draw 7 hundred-tiles, 2 ten-tiles, and 6 one-tiles in your notebook.

Given number: 504

We break 504 into hundreds, tens, and ones:
$$504 = 500 + 0 + 4$$

- Hundreds: 5 big tiles
- Tens: 0 medium tiles (none)
- Ones: 4 small tiles

Draw 5 hundred-tiles and 4 one-tiles in your notebook. There are no ten-tiles.

Given number: 620

We break 620 into hundreds, tens, and ones:
$$620 = 600 + 20 + 0$$

- Hundreds: 6 big tiles
- Tens: 2 medium tiles
- Ones: 0 small tiles (none)

Draw 6 hundred-tiles and 2 ten-tiles in your notebook. There are no one-tiles.

Given number: 700

We break 700 into hundreds, tens, and ones:
$$700 = 700 + 0 + 0$$

- Hundreds: 7 big tiles
- Tens: 0 medium tiles (none)
- Ones: 0 small tiles (none)

Draw 7 hundred-tiles in your notebook. There are no ten-tiles or one-tiles.

We place each number on the number line between 500 and 700.

- $530$ lies between 500 and 600, closer to 500. It is 3 tens after 500.
- $540$ lies between 500 and 600. It is 4 tens after 500.
- $590$ lies between 500 and 600, very close to 600. It is 9 tens after 500.
- $628$ lies between 600 and 700, closer to 600. It is 2 tens and 8 ones after 600.
- $696$ lies between 600 and 700, very close to 700. It is 9 tens and 6 ones after 600.

On the number line, mark these points in order: $530, 540, 590, 628, 696$.

We place each number on the number line between 700 and 900.

- $703$ lies just after 700 (3 ones after 700).
- $721$ lies between 700 and 800, closer to 700 (2 tens and 1 one after 700).
- $759$ lies between 700 and 800, closer to 800 (5 tens and 9 ones after 700).
- $810$ lies between 800 and 900, closer to 800 (1 ten after 800).
- $855$ lies between 800 and 900, in the middle (5 tens and 5 ones after 800).
- $887$ lies between 800 and 900, closer to 900 (8 tens and 7 ones after 800).

On the number line, mark these points in order: $703, 721, 759, 810, 855, 887$.

We find the neighbouring hundreds (multiples of 100), neighbouring fifties (multiples of 50), and neighbouring tens (multiples of 10) for each number.

183:
- Neighbouring Hundreds: $100$ and $200$ (since 100 < 183 < 200)
- Neighbouring Fifties: $150$ and $200$ (since 150 < 183 < 200)
- Neighbouring Tens: $180$ and $190$ (since 180 < 183 < 190)

345:
- Neighbouring Hundreds: $300$ and $400$ (since 300 < 345 < 400)
- Neighbouring Fifties: $300$ and $350$ (since 300 < 345 < 350)
- Neighbouring Tens: $340$ and $350$ (since 340 < 345 < 350)

693:
- Neighbouring Hundreds: $600$ and $700$ (since 600 < 693 < 700)
- Neighbouring Fifties: $650$ and $700$ (since 650 < 693 < 700)
- Neighbouring Tens: $690$ and $700$ (since 690 < 693 < 700)

734:
- Neighbouring Hundreds: $700$ and $800$ (since 700 < 734 < 800)
- Neighbouring Fifties: $700$ and $750$ (since 700 < 734 < 750)
- Neighbouring Tens: $730$ and $740$ (since 730 < 734 < 740)

899:
- Neighbouring Hundreds: $800$ and $900$ (since 800 < 899 < 900)
- Neighbouring Fifties: $850$ and $900$ (since 850 < 899 < 900)
- Neighbouring Tens: $890$ and $900$ (since 890 < 899 < 900)

Completed Table:

| Number | Neighbouring Hundreds | Neighbouring Fifties | Neighbouring Tens |
|---|---|---|---|
| 468 | 400 and 500 | 450 and 500 | 460 and 470 |
| 183 | 100 and 200 | 150 and 200 | 180 and 190 |
| 345 | 300 and 400 | 300 and 350 | 340 and 350 |
| 693 | 600 and 700 | 650 and 700 | 690 and 700 |
| 734 | 700 and 800 | 700 and 750 | 730 and 740 |
| 899 | 800 and 900 | 850 and 900 | 890 and 900 |

Crane 1 — To reach the worm (going left/backward):

Given: $667 - ? = 650$

$$667 - 650 = 17$$

So the crane takes 17 steps (each step of length 1), or it can take 1 step of 10 and 1 step of 7, or 1 step of 17.

Answer: $667 - 17 = 650$. Number of steps: 17 (or 2 jumps: first 10, then 7).

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Crane 1 — To reach the fish (going right/forward):

Given: $667 + ? = 700$

$$700 - 667 = 33$$

So the crane takes 33 steps, or it can take 1 step of 30 and 1 step of 3, or 1 step of 33.

Answer: $667 + 33 = 700$. Number of steps: 33 (or 2 jumps: first 30, then 3).

---

Crane 2 — To reach the worm (going left/backward):

The crane is standing at some number and needs to reach 750. Based on the number line context (the second crane appears to be near 780):

Given: $780 - ? = 750$

$$780 - 750 = 30$$

Answer: $780 - 30 = 750$. Number of steps: 30.

*(Note: The exact starting position of Crane 2 depends on the figure. If the crane stands at 780, then 30 steps back reaches 750.)*

---

Crane 2 — To reach the fish (going right/forward):

Given: $780 + ? = 800$

$$800 - 780 = 20$$

Answer: $780 + 20 = 800$. Number of steps: 20.

Let us apply each clue to the grid:

Clue 1: 597
- Strike out: 597 ✓

Clue 2: A number with digit 4
- Numbers in grid containing digit 4: 574, 614
- Strike out: 574, 614 ✓

Clue 3: Numbers between 595 and 605
- Numbers strictly between 595 and 605: 596, 597, 598, 599, 600, 601, 602, 603, 604
- Numbers in grid: 599, 597 (already struck), 600, 602
- Strike out: 599, 600, 602 ✓ (597 already done)

Clue 4: A number with 1 as the tens digit
- Tens digit = 1 means numbers like _1_: 610–619
- Numbers in grid: 616, 614 (already struck)
- Strike out: 616 ✓

Clue 5: Two more than 610
- $610 + 2 = 612$
- 612 is not in the given grid, so no number to strike out from the grid for this clue.

Clue 6: 5 less than 625
- $625 - 5 = 620$
- 620 is not in the given grid, so no number to strike out from the grid for this clue.

Summary of numbers struck out from the grid:
- Clue 1: 597
- Clue 2: 574, 614
- Clue 3: 599, 600, 602
- Clue 4: 616

Total numbers struck out: 7 (597, 574, 614, 599, 600, 602, 616)

Note: The exact centre numbers depend on the figures which are not fully visible. The method is shown below using examples.

Example for 456:
$$456 = 400 + 50 + 6$$
$$456 = 400 + 56$$
$$456 = 450 + 6$$
$$456 = 460 - 4$$
$$456 = 500 - 44$$

Example for 321:
$$321 = 300 + 20 + 1$$
$$321 = 300 + 21$$
$$321 = 320 + 1$$
$$321 = 330 - 9$$
$$321 = 400 - 79$$

Write as many number sentences as you can for the number given in the centre of your figure.

Note: The exact conditions depend on the figures. The general method is:

- Read the condition carefully (e.g., 'greater than 500 and less than 600').
- Think of numbers that satisfy all the given conditions.
- Write those numbers in the blank spaces.

Example: If the condition is: greater than 400, less than 500, has 3 in the tens place:
- Possible answers: 430, 431, 432, 433, 434, 435, 436, 437, 438, 439

Fill in numbers that satisfy all the conditions shown in your figure.

Note: The exact conditions in the figure are not fully visible, but the standard approach for such puzzles is:

Given numbers: 384, 23, 176, 905

Arrange them from smallest to greatest:
23 < 176 < 384 < 905

Typical conditions in such puzzles:
- Largest number: 905
- Smallest number: 23
- Numbers in between (smaller to greater): 176, 384

If the puzzle asks to place them so that each circle satisfies a condition (e.g., greater than the one before), place them as:
$$23 \rightarrow 176 \rightarrow 384 \rightarrow 905$$

Fill the circles according to the conditions shown in your figure using this ordering.

We match each quantity with the most appropriate range:

- Number of children in your school → 501–1000 (A school usually has several hundred children)
- Number of books in your classroom → 11–50 (A classroom typically has 20–40 books)
- Number of people in a bus → 51–100 (A bus carries about 50–80 people)
- Number of pages in your mathematics book → 101–200 (A Class 3 Maths book has around 100–150 pages)
- Number of steps you walk in a day → More than 1000 (We walk thousands of steps in a day)
- Number of stars in the sky → More than 1000 (There are countless stars)
- Number of flowers in a garland → 51–100 (A garland typically has 50–100 flowers)

Note: Some quantities can match more than one range depending on the context. Discuss with your teacher.

Let us work out each description:

1. 'I have 2 zeroes as digits and am very close to 99'
- A number close to 99 with two zeroes: 100 (digits are 1, 0, 0 — two zeroes, and 100 is just 1 more than 99)
- Matches: 100 ✓

2. 'I have 3 hundreds, 6 tens and 7 ones'
- $3 \times 100 + 6 \times 10 + 7 \times 1 = 300 + 60 + 7 = 367$
- Matches: 367 ✓

3. 'I have zero tens and zero ones'
- A number with 0 tens and 0 ones is a multiple of 100, e.g., 400.
- Matches: 400 ✓

4. 'I am century + half century'
- Century = 100, Half century = 50
- $100 + 50 = 150$
- Matches: 150 ✓

5. 'I come between 400 and 450, and I have 5 as a digit'
- Numbers between 400 and 450 with digit 5: 405, 415, 425, 435, 445
- From the given options: 425 ✓

Final Matching:
- I have 2 zeroes as digits and am very close to 99 → 100
- I have 3 hundreds, 6 tens and 7 ones → 367
- I have zero tens and zero ones → 400
- I am century + half century → 150
- I come between 400 and 450, and I have 5 as a digit → 425

The hundreds are multiples of 100:
$$100, 200, 300, 400, 500, 600, 700, 800, 900, 1000$$

There are 10 hundreds from 100 to 1000.

Observation:
- $123$: digits are 1, 2, 3 — three consecutive digits in increasing order.
- $345$: digits are 3, 4, 5 — three consecutive digits in increasing order.
- $789$: digits are 7, 8, 9 — three consecutive digits in increasing order.
- $876$: digits are 8, 7, 6 — three consecutive digits in decreasing order.
- $321$: digits are 3, 2, 1 — three consecutive digits in decreasing order.

These numbers have consecutive digits (either going up or going down).

More such numbers (increasing): $234, 456, 567, 678$
More such numbers (decreasing): $987, 765, 654, 543, 432$

Twin digit numbers (2-digit, both digits same):
$$11, 22, 33, 44, 55, 66, 77, 88, 99$$

Triplet digit numbers (3-digit, all digits same):
$$111, 222, 333, 444, 555, 666, 777, 888, 999$$

These are numbers where all digits are the same.

These numbers are called palindromes — they read the same forwards and backwards.

Examples of 3-digit palindromes:
$$121, 131, 141, 151, 161, 171, 181, 191$$
$$202, 212, 232, 242, 252, 262, 272, 282, 292$$
$$303, 313, 323, 333, 343, 363, 373, 383, 393$$
$$404, 414, 424, 434, 444, 454, 464, 474, 484, 494$$
$$505, 515, 525, 535, 545, 555, 565, 575, 585, 595$$
$$606, 616, 626, 636, 646, 656, 676, 686, 696$$
$$707, 717, 727, 737, 747, 757, 767, 777, 787, 797$$
$$808, 818, 828, 838, 848, 858, 878, 888, 898$$
$$909, 919, 929, 939, 949, 959, 969, 979, 989, 999$$

The pattern: a 3-digit palindrome has the form $ABA$ where the first and last digit are the same.

Numbers with at least one zero:

Numbers with one zero:
$$102, 103, 104, 105, 201, 203, 301, 302, 401, 501, 601, 701, 801, 901$$
$$120, 130, 140, 150, 160, 170, 180, 190, 230, 240, 310, 320, 410, 510, 610, 710, 810, 910$$

Numbers with two zeroes:
$$100, 200, 300, 400, 500, 600, 700, 800, 900$$
$$101, 202, 303, 404, 505, 606, 707, 808, 909$$

Some examples: $100, 200, 300, 400, 500, 600, 700, 800, 900, 101, 201, 301, 401, 501, 601, 701, 801, 901, 110, 120, 130, 210, 220, 304, 405, 506$

We have 6 slips. Each slip can be labelled 100, 10, or 1.

The total value = (number of 100-slips × 100) + (number of 10-slips × 10) + (number of 1-slips × 1)

Let the number of 100-slips = $a$, 10-slips = $b$, 1-slips = $c$, where $a + b + c = 6$.

Some examples of numbers we can make:

| 100-slips | 10-slips | 1-slips | Number |
|---|---|---|---|
| 6 | 0 | 0 | 600 |
| 5 | 1 | 0 | 510 |
| 5 | 0 | 1 | 501 |
| 4 | 2 | 0 | 420 |
| 4 | 1 | 1 | 411 |
| 4 | 0 | 2 | 402 |
| 3 | 3 | 0 | 330 |
| 3 | 2 | 1 | 321 |
| 3 | 1 | 2 | 312 |
| 3 | 0 | 3 | 303 |
| 2 | 4 | 0 | 240 |
| 2 | 3 | 1 | 231 |
| 2 | 2 | 2 | 222 |
| 2 | 1 | 3 | 213 |
| 2 | 0 | 4 | 204 |
| 1 | 5 | 0 | 150 |
| 1 | 4 | 1 | 141 |
| 1 | 3 | 2 | 132 |
| 1 | 2 | 3 | 123 |
| 1 | 1 | 4 | 114 |
| 1 | 0 | 5 | 105 |
| 0 | 6 | 0 | 60 |
| 0 | 5 | 1 | 51 |
| 0 | 4 | 2 | 42 |
| 0 | 3 | 3 | 33 |
| 0 | 2 | 4 | 24 |
| 0 | 1 | 5 | 15 |
| 0 | 0 | 6 | 6 |

Many different numbers can be made!

Largest number:
To make the largest number, put 100 on all 6 slips.
$$6 \times 100 = 600$$
Largest number = 600

Smallest number:
To make the smallest number, put 1 on all 6 slips.
$$6 \times 1 = 6$$
Smallest number = 6

Numbers that cannot be made:
The total value must be a combination of 100s, 10s, and 1s using exactly 6 slips.

For example:
- We cannot make 1 (we need at least 6 slips, so minimum value is 6).
- We cannot make 2, 3, 4, or 5.
- We cannot make 601 (that would need more than 6 slips: six 100s + one 1 = 7 slips).
- Numbers greater than 600 cannot be made.
- Numbers between 6 and 600 that cannot be formed with exactly 6 slips also cannot be made.

For example, 7 cannot be made: to get 7 ones we need 7 slips, but we only have 6.

So numbers less than 6 and numbers greater than 600 cannot be made.

Using digits 3 and 8 (repetition allowed):

2-digit numbers:
$$33, 38, 83, 88$$

3-digit numbers:
$$333, 338, 383, 388, 833, 838, 883, 888$$

Arrange 2-digit numbers from smaller to greater:
33 < 38 < 83 < 88

Arrange 3-digit numbers from smaller to greater:
333 < 338 < 383 < 388 < 833 < 838 < 883 < 888

Note: The exact words on the cards depend on the figures. The Bhutasankhya system uses words from nature/culture to represent digits. For example, 'eyes' = 2, 'moon' = 1, 'fire' = 3, etc.

Based on the examples given in the chapter (Teji shows blue cards = 12, yellow cards = 14):

- The number of letters/syllables in each word gives the digit value.
- Read the digits from right to left (as is traditional in Bhutasankhya).

(a) Read the words on the cards, find the digit each word represents, then write the number by reading digits from right to left.

(b) Similarly, read the words, find the digits, and write the number.

Example method: If card (a) shows words representing digits 3 and 5 (reading right to left gives 53), then the number is 53.

Apply the same method to the words shown on your cards.

In Bhutasankhya, words are chosen so that the number of letters (or a cultural association) gives the digit.

Example words for digits 0–9:
- 0 → sky (शून्य/zero)
- 1 → moon (there is 1 moon)
- 2 → eyes (we have 2 eyes)
- 3 → fire (तीन/three sacred fires)
- 4 → directions (4 directions)
- 5 → fingers (5 fingers on one hand)
- 6 → seasons (6 seasons in Indian calendar)
- 7 → days (7 days in a week)
- 8 → directions (8 directions)
- 9 → planets (9 planets/navagraha)

Making cards for 15:
- 15 → digits are 1 and 5
- In Bhutasankhya, write right to left: first card = word for 5, second card = word for 1
- Cards: [fingers] [moon] → read right to left = 15

Making cards for 27:
- 27 → digits are 2 and 7
- Cards: [days] [eyes] → read right to left = 27

Making cards for 94:
- 94 → digits are 9 and 4
- Cards: [directions(4)] [planets(9)] → read right to left = 94

You can choose your own words from your local language and culture!