House of Hundreds - II — NCERT Solutions

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:

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!