3. Number Play

• Reading the time on a clock
• Dates on a calendar
• Counting money while shopping
• Marks in a test
• Measuring height and weight

No. A child at the end has only one neighbour. So that child can say only 0 or 1.

Only if all the children have the same height.

If the heights are different, the shortest child will always have a taller neighbour. So not everyone can say 0.

Yes. For example, in the first picture two children standing next to each other both say 1.

Yes. Let them stand in increasing order of height, from shortest to tallest.

Each of the first four has one taller neighbour, so they say 1. The tallest child at the end says 0.

No. The tallest child has no taller neighbour. So the tallest child must say 0.

Yes. Let the heights (in order) be: tallest, 3rd tallest, shortest, 4th tallest, 2nd tallest.

For example, heights 150, 130, 110, 120, 140 cm give 0, 1, 2, 1, 0.

Make the 2nd and 4th children the two shortest. Each of them has two taller neighbours.

So at most 2 children can say ‘2’.

Supercells: 6828 9435 8000

The supercells are 6000, 1258 and 9700 (the coloured cells). Other answers are also possible.

Put a big number in every alternate cell, starting from the first cell.

5 supercells.

Pattern: For an even number of cells, the maximum is half the number of cells. For an odd number of cells, it is half of (number of cells + 1).

Method: Start with a supercell in the first cell, and make every alternate cell a supercell.

No. The cell with the largest number is always bigger than its neighbours. So there is always at least one supercell.

The largest number is always a supercell, because it is bigger than all its neighbours.

The smallest number can never be a supercell, because its neighbours are bigger than it.

8 is the second largest number. It is next to 9, so it is not a supercell.

Yes. 2 (second smallest) is at the end next to 1, so it is a supercell. 8 (second largest) is next to 9, so it is not a supercell.

• Fill 9 cells so that there are exactly 4 supercells.
• Fill the table so that only the even numbers are supercells.
• Fill 9 cells so that the smallest supercell is as small as possible.

One possible answer (supercells in bold):

The biggest number in the table is 96,310

The smallest even number in the table is 10,396

The smallest number greater than 50,000 in the table is 60,193

Between 1000 and 2000: 1050, 1500

Between 2000 and 3000: 2180, 2754

Between 3000 and 4000: 3050, 3600

Between 5000 and 6000: 5030, 5300

Between 8000 and 9000: 8400

Between 9000 and 10,000: 9590, 9950

(a) Gap of 5: 1990, 1995, 2000, 2005, 2010, 2015, 2020, 2025, 2030, 2035

(b) Gap of 1: 9993, 9994, 9995, 9996, 9997, 9998, 9999, 10,000, 10,001, 10,002

(c) Gap of 1: 15,077, 15,078, 15,079, 15,080, 15,081, 15,082, 15,083, 15,084, 15,085, 15,086

(d) Gap of 1000: 83,705, 84,705, 85,705, 86,705, 87,705, 88,705, 89,705, 90,705, 91,705, 92,705

(a) 59, 77, 86, 95, 149, 248, 356, 2345, 11,237

(b) Smallest number = 59

(c) Largest 5-digit number = 95,000

(d) There is no biggest number. We can put more zeros: 9500, 95,000, 9,50,000, 95,00,000, … Each one is bigger and still has digit sum 14.

40 to 49: 4, 5, 6, 7, 8, 9, 10, 11, 12, 13

50 to 59: 5, 6, 7, 8, 9, 10, 11, 12, 13, 14

60 to 69: 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

70: 7

• Within each group of ten, the digit sum increases by 1.
• When we move from 49 to 50, the digit sum drops from 13 to 5.
• The same digit sum appears in different groups, for example 49, 58 and 67 all have digit sum 13.

123 → 6, 234 → 9, 345 → 12, 456 → 15, 567 → 18, 678 → 21, 789 → 24

Pattern: The digit sums go up by 3 each time. Each sum is 3 times the middle digit.

The pattern stops after 789, because there is no digit after 9.

1–100: 7 in the ones place 10 times (7, 17, …, 97) + 7 in the tens place 10 times (70–79) = 20 times

1–1000: 100 times in each of the ones, tens and hundreds places = 300 times

111, 121, 131, 212, 222, 232, 313, 323, 333 — there are 9 palindromes.

Yes. For every 2-digit number we finally reach a palindrome. Example: 39 + 93 = 132, 132 + 231 = 363.

Some numbers take many steps. For example, 89 takes 24 steps.

The ones digit must be odd. If u = 1, then t = 2 and h = 4. (If u = 3, h becomes 12, which is not a digit.)

The number is 12,421 — twelve thousand four hundred twenty-one.

Example: 321 → 321 − 123 = 198 → 981 − 189 = 792 → 972 − 279 = 693 → 963 − 369 = 594 → 954 − 459 = 495 → 954 − 459 = 495 …

The number 495 starts repeating.

All digits same: 1:11, 2:22, 3:33, 4:44, 5:55, 11:11

Hour repeated as minutes: 10:10, 11:11, 12:12 (and 1:01, 2:02, …, 9:09)

Palindromic times: 10:01, 11:11, 12:21, and for each hour 1 to 9: h:0h, h:1h, h:2h, h:3h, h:4h, h:5h (for example 1:01, 1:11, 1:21, 1:31, 1:41, 1:51).

20/01/2001, 20/02/2002, 20/04/2004, 20/08/2008, 20/11/2011, 19/11/1911, 19/12/1912

Dates of this form since the year 2000:

10/02/2001, 20/02/2002, 01/02/2010, 11/02/2011, 21/02/2012, 02/02/2020, 12/02/2021, 22/02/2022

Yes. The calendar repeats when both years start on the same day and both are leap years or both are not leap years.

A non-leap year repeats after 6 or 11 years. For example, the calendar of 2026 will be used again in 2037.

A leap year usually repeats after 28 years. For example, 2024 repeats in 2052.

(a) Digits 7, 4, 3, 1: 7431 − 1347 = 6084 (more than 5085)

(b) Digits 7, 4, 3, 3: 7433 − 3347 = 4086 (less than 5085)

(c) Digits 7, 4, 3, 3: 7433 + 3347 = 10,780 (more than 9779)

(d) Digits 7, 4, 3, 1: 7431 + 1347 = 8778 (less than 9779)

Smallest 5-digit palindrome = 10,001; largest = 99,999

Sum = 99,999 + 10,001 = 1,10,000

Difference = 99,999 − 10,001 = 89,998

Next palindromic time = 11:11. From 10:01 to 11:11 = 1 hour 10 minutes = 70 minutes

The one after that = 12:21. That is 70 minutes after 11:11 (140 minutes after 10:01).

It takes 7 rounds.

28,000 = 25,000 + 1,500 + 1,500

61,600 = 60,000 + 400 × 4

31,000 = 25,000 + 1,500 × 4

63,000 = 60,000 + 1,500 × 2

19,500 = 13,000 + 1,500 × 3 + 400 × 5

20,900 = 13,000 + 1,500 × 5 + 400

1,000 cannot be made. Only 400 is smaller than 1,000, and 400 + 400 = 800, 400 + 400 + 400 = 1,200.

14,000 = 1,500 × 8 + 400 × 5 = 12,000 + 2,000

15,000 = 13,000 + 400 × 5 = 15,000

16,000 = 13,000 + 1,500 × 2 = 16,000

Only 1,000 cannot be made. All other thousands can be made.

45,000 = 40,000 + 7,000 − 1,500 − 800 + 300

5,900 = 7,000 − 800 − 300

17,500 = 12,000 + 7,000 − 1,500

21,400 = 12,000 + 7,000 + 1,500 + 300 + 300 + 300

(a) 5-digit + 5-digit gives a 5-digit number — Sometimes true. 20,000 + 30,000 = 50,000, but 60,000 + 40,000 = 1,00,000.

(b) 4-digit + 2-digit gives a 4-digit number — Sometimes true. 2000 + 50 = 2050, but 9990 + 50 = 10,040.

(c) 4-digit + 2-digit gives a 6-digit number — Never true. The biggest sum is 9999 + 99 = 10,098.

(d) 5-digit − 5-digit gives a 5-digit number — Sometimes true. 50,000 − 10,000 = 40,000, but 12,000 − 10,000 = 2000.

(e) 5-digit − 2-digit gives a 3-digit number — Never true. The smallest answer is 10,000 − 99 = 9901.

(a) 12 × 40 + 10 × 50 = 480 + 500 = 980

(b) 20 cells with 5 dots + 44 cells with 1 dot = 100 + 44 = 144

(c) 32 × 32 + 16 × 64 = 1024 + 1024 = 2048

(d) 17 cells with 3 dots + 18 cells with 4 dots = 51 + 72 = 123

(e) 7 hexagons (2 × 210 + 3 × 150 + 2 × 90 = 1050) + 6 corner pieces (2 × 60 + 2 × 100 + 2 × 140 = 600) = 1650

(f) 1000 + 4 × 500 + 8 × 250 + 18 × 125 = 1000 + 2000 + 2000 + 2250 = 7250

7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1

28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1

Yes, every number we try reaches 1. But no one has proved that it happens for every number. So it is still a conjecture.

1. Number of words in your maths textbook: more than 5000 (the book has many pages with many words on each page).

2. Students who come by bus: depends on your school.

3. Roshan’s estimate of ₹100: No. Milk and three types of fruit for 5 people will cost more than ₹100.

4. Gandhinagar to Kohima: about 2500 km.

5. Sheetal’s 13,000 hours: No. About 6 hours a day × 200 days a year × 7 years ≈ 8400 hours. 13,000 hours is too high.

The first player can always win.

The first player should say 1, then 5, 9, 13, 17 and 21. These numbers increase by 4.

The second player can always win.

The second player should always say 11, 22, 33, 44, 55, 66, 77, 88 and 99 (multiples of 11).

Swap the digits 6 and 1 in 62,871 to get 12,876.

Now 39,344, 23,609, 45,306 and 50,319 are supercells.

Example: 2014 → 4210 − 0124 = 4086 → 8640 − 0468 = 8172 → 8721 − 1278 = 7443 → 7443 − 3447 = 3996 → 9963 − 3699 = 6264 → 6642 − 2466 = 4176 → 7641 − 1467 = 6174.

2014 takes 7 rounds. Try it for your own year of birth.

Largest = 73,999

Smallest = 35,111

Closest to 50,000 = 51,111

(51,111 is 1,111 away from 50,000, but 39,999 is 10,001 away.)

About 52 Sundays + 15 festival holidays + 50 days of vacation ≈ 117 days, so about 120 days. Check the school calendar for the exact number.

Mug: about 1 litre. Bucket: about 15–20 litres. Overhead tank: about 500–1000 litres.

18,000 + 300 + 370 = 18,670

Another answer: 17,670 + 500 + 500 = 18,670

Number chosen: 250. Pattern: a 5 × 5 square of 10s: 25 × 10 = 250.

Another: 300 = 6 rows of five 10s (30 × 10 = 300).

A power of 2 is even, so we take half. Half of a power of 2 is again a power of 2.

We keep halving: … 16, 8, 4, 2, 1. So we always reach 1.

100, 50, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1

Yes, it reaches 1.

Work backwards: 22, 18, 14, 10, 6, 2 (subtract 4 each time).

The first player should start by saying 2, and then always say 6, 10, 14, 18 and 22.