What is the number of digits in the number obtained by placing all the numbers from 1 to 123

Start with the sequence of non-zero digits 123456789. The problem is to place plus or minus signs between them so that the result of thus described arithmetic operation will be 100.

We got one answer

12 + 3 - 4 + 5 + 67 + 8 + 9 = 100

and suggested there existed at least one more. I do not claim to have made an exhaustive search but there appear to be more than just two answers. One of this is

123 + 4 - 5 + 67 - 89 = 100

I am sure there it at least another one. Want to find it?

There is a keen observation that in the two examples above at least one of the operations is subtraction. And this is also true for all additive (the ones in which the only allowed operations are addition and subtraction) examples below. In fact, it is impossible to avoid subtraction even if the digits come in an arbitrary order. To figure out why, it may be helpful to recollect the notion of digital roots.

You may allow for operations other than addition and subtraction. This leads to a completely new set of problems with numbers having fractional parts. Variations include setting targets other than 100. Here, for example, a representation of one that uses all ten digits:

1 = 148/296 + 35/70

There are many ways to merrily spend time solving arithmetic problems. One way is to attempt representing numbers with limited means. For example, I can represent 100 with five threes as 100 = 33×3 + 3/3. It's surprising how many numbers could be represented this way.

In the 1960s another kind of number puzzles have become very popular. Cryptarithms are brain teasers obtained when digits in numerical calculations have been replaced by letters. Customarily, distinct letters stand for different digits. Stars substitute for any digit and are not related to each other.

I received the following letter from Belgium:

From: Gui et Nicole RULMONT
Date: Tue, 22 Apr 1997 17:02:44 +0200

Dear Cut-the-Knot,

First please excuse my English. I am Belgian and I am very interested by your site!

You wrote in "Fun with digits": Start with the sequence of non-zero digits 123456789. The problem is to place plus or minus signs between them so that the result of thus described arithmetic operation will be 100.

Some years ago, il found in the french magazine Science et Vie the 11 solutions:

1 + 2 + 34 - 5 + 67 - 8 + 9 = 100 12 + 3 - 4 + 5 + 67 + 8 + 9 = 100 123 - 4 - 5 - 6 - 7 + 8 - 9 = 100 123 + 4 - 5 + 67 - 89 = 100 123 + 45 - 67 + 8 - 9 = 100 123 - 45 - 67 + 89 = 100 12 - 3 - 4 + 5 - 6 + 7 + 89 = 100 12 + 3 + 4 + 5 - 6 - 7 + 89 = 100 1 + 23 - 4 + 5 + 6 + 78 - 9 = 100 1 + 23 - 4 + 56 + 7 + 8 + 9 = 100

1 + 2 + 3 - 4 + 5 + 6 + 78 + 9 = 100

If we put a "-" before 1, we have one more solution:

-1 + 2-3 + 4 + 5 + 6 + 78 + 9 = 100

Using the "." decimal separation I found another solution:

1 + 2.3 - 4 + 5 + 6.7 + 89 = 100 (solution of my own)

What about 987654321 ? There are 15 solutions, said Science et Vie:

98 - 76 + 54 + 3 + 21 = 100 9 - 8 + 76 + 54 - 32 + 1 = 100 98 + 7 + 6 - 5 - 4 - 3 + 2 - 1 = 100 98 - 7 - 6 - 5 - 4 + 3 + 21 = 100 9 - 8 + 76 - 5 + 4 + 3 + 21 = 100 98 - 7 + 6 + 5 + 4 - 3 - 2 - 1 = 100 98 + 7 - 6 + 5 - 4 + 3 - 2 - 1 = 100 98 + 7 - 6 + 5 - 4 - 3 + 2 + 1 = 100 98 - 7 + 6 + 5 - 4 + 3 - 2 + 1 = 100 98 - 7 + 6 - 5 + 4 + 3 + 2 - 1 = 100 98 + 7 - 6 - 5 + 4 + 3 - 2 + 1 = 100 98 - 7 - 6 + 5 + 4 + 3 + 2 + 1 = 100 9 + 8 + 76 + 5 + 4 - 3 + 2 - 1 = 100 9 + 8 + 76 + 5 - 4 + 3 + 2 + 1 = 100

9 - 8 + 7 + 65 - 4 + 32 - 1 = 100

Write the sign "-", three solutions:

-9 + 8 + 76 + 5-4 + 3 + 21 = 100 -9 + 8 + 7 + 65 - 4 + 32 + 1 = 100

-9-8 + 76 - 5 + 43 + 2 + 1 = 100

With the decimal point:</>

9 + 87.6 + 5.4 - 3 + 2 - 1 = 100 (solution of my own)

If I "shuffle" the digits there are many solutions. I found some when I was young, for example:

91 + 7.68 + 5.32 - 4 = 100 98.3 + 6.4 - 5.7 + 2 - 1 = 100 538 + 7 - 429 - 13 = 100

(8×9.125) + 37 - 6 - 4 = 100 etc etc etc ....

very interested by cryptarithms and I collect them. Do you want to receive french cryptarithms ? Do you know non-english cryptarithms ? Thanks!

Gui et Nicole Rulmont

Anthony Lesar notes that the solution 1 + 2 + 3 - 4 + 5 + 6 + 78 + 9 = 100 could be a little modified without changing the result: 1! + 2! + 3 - 4 + 5 + 6 + 78 + 9 = 100.

Note: There is a whole bunch of pages that offer practice problems of this kind. Also, Inder Jeet Taneja has gathered a fantastic collection of various sequential representations of numbers from 1 to 11111.

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3-digit numbers begin with 100 and end on 999. These numbers consist of 3 digits in which the first digit should be 1 or greater than 1 and the remaining 3 digits can be any number from 0 to 9. Learning 3-digit numbers is the building block for higher-digit numbers. Let us explore more about the importance, formation, and place value of numbers up to 3 digit.

What are 3-Digit Numbers?

3-digit numbers are those numbers that consist of only 3 digits. They start from 100 and go on till 999. For example, 673, 104, 985 are 3-digit numbers. It is to be noted that the first digit of a three-digit number cannot be zero because in that case, it becomes a 2-digit number. For example, 045 becomes 45.

Place Value of 3-Digit Numbers

Every three-digit number’s value can be found by checking the place value of each digit. Let us consider the number 243. The first digit at the rightmost position is said to be at units place, so it will be multiplied by 1. Hence, the product is 3 × 1 = 3. Then the second number is 4, and because it is at tens place, it is multiplied by 10. The value, therefore, is 4 × 10 = 40. The third number 2 is at the hundreds place. So 2 is multiplied by 100 and its value is 2 × 100 = 200. Therefore the number is 200 + 40 + 3 = 243.

Decomposing a 3-digit number: In a three-digit number, there are three place values used – hundreds, tens, and units. Let us take one example to understand it better. Here, 465 is a three-digit number and it is decomposed in the form of a sum of three numbers. As 5 is on the units place, 60 is on the tens place and 400 is on the hundreds place.

Significance of Zero in 3-digit numbers: The number zero does not make any contribution to a 3-digit number if it is placed in a position where there are no other non-zero numbers to its left. So how is 303 different from 033 or even 003? In 033, the values are (0 × 100) + (3 × 10) + (3 × 1) = 0 + 30 + 3= 33 which means that the number actually becomes a 2-digit number, i.e., 33, or in the case of 003, it becomes a single-digit number, i.e., 3. In these two examples, the zero does not contribute any value to the number, so the numbers can be expressed as 33 or 3 as well.

Expanded Form of 3-Digit Numbers

The expanded form of a 3-digit number can be expressed and written in three different ways. Consider a three-digit number 457. The number 457 can be written in one form as 457 = (4 × hundreds) + (5 × tens) + (7 × ones). In the second way, the number 457 can be written as 457 = (4 × 100) + (5 × 10) + (7 × 1). And finally the number 457 can be expanded in the form as 457 = 400 + 50 + 7. All the three ways of writing numbers in the expanded form are correct. Writing a 3-digit number in the expanded form helps to know the constituents of the number.

Basically splitting or expanding a 3-digit number helps us to understand more about the 3-digit number. By splitting we know the number of hundreds, tens, and units available in the 3-digit number.

Important Notes on 3-digit Numbers

• 100 is the smallest 3-digit number and 999 is the greatest 3-digit number.
• A 3-digit number cannot start with 0.
• 10 tens make 1 hundred which is the smallest 3-digit number and 10 hundred make a thousand which is the smallest 4-digit number.
• A 3-digit number can also have two zeros but the two zeros should be in the tens place and the units place, for example, 100, 200, 300, 400. It is to be noted that the zeros cannot be in the hundreds place because in that case it becomes a 2-digit number. For example, 067 becomes 67.

Common Mistakes of Numbers up to 3-Digits

Some of the common mistakes are observed while writing or reading a 3-digit number. These mistake in reading and interpreting a 3-digit number is often understood as some other number. In the process of reading, writing, and interpreting a 3-digit number, the place value of the digits should be rightly interpreted. Here we have listed below the three common mistakes often committed by children in writing three-digit numbers.

• Misconception 1: Children make mistakes identifying numbers when there is a zero in the units place or tens place. Example: When asked to read 130 and 103, students may get confused. It helps them to model the numbers through Base-10 blocks. That way they can see the ten’s and one’s place value explicitly.
• Misconception 2: When asked to write “one hundred twenty-three," students often write 100 first and then attach 23 to it thus ending up with the number “10023” Fact: This misconception arises due to a superficial understanding of place values. Using the base-10 blocks or abacus show children that a digit has different values based on its position.
• Misconception 3: Sometimes when asked to form the smallest 3-digit number given three digits that include zero, children place the zero in the left-most position. Fact: This is incorrect. Zero cannot be in the hundreds place if we are creating a 3-digit number. For example: the smallest 3-digit number using all digits of 5, 0, and 7 is 507 and not 057

Operations on Numbers up to 3-Digits

The four arithmetic operations of addition, subtraction, multiplication, and division can be conveniently performed across 3-digit numbers. In the process of performing these arithmetic operations, the place value of the corresponding number should be rightly matched. An error in matching the place value could result in wrong answers. Here we shall look at a simple activity using 3 digit numbers, to help us understand the changing pattern in each of the digits of the hundredth place, ten's place, and unit's place. This activity shall help in a better understanding of the learning needed for the 3 digit numbers.

• Get students to skip count by 10 and 100 to build fluency with 3-digit numbers. First, start at 100. Then start from any random 3-digit number like 136.

• Help children spot the pattern that when skip counting by 10, the digit in the ones place value does not change. Similarly, when skip counting by 100, the digits in the ones place and tens place does not change.

• Use a 100-square grid to build fluency. Let students spot the pattern that moving one row up or down is the same as skip counting by 10. Moving columns (left or right) increases or decreases numbers by 1.

• Often children are given three digits and asked to find the largest and smallest number three-digit number using all digits. The trick here is to arrange all digits in descending order to find the largest number.
  To find the smallest number, arrange all digits in ascending order. But keep in mind that if zero is one of the digits, it cannot be placed to the left. E.g. Using the digits 7, 3, and 6, the largest number is 763 (digits in descending order) and the smallest number is 367 (digits in ascending order). Using the digits 4, 0, and 8, the largest number will be 840 but the smallest 3-digit number is 408 and not 048.

Smallest 3-Digit Number

The smallest 3-digit number is 100 because its predecessor is 99 which is a two-digit number. 3-digit numbers start from 100 and end on 999.

Greatest 3-Digit Number

The greatest 3-digit number is 999 because its successor is 1000 which is a four-digit number. 3-digit numbers start from 100 and end on 999.

☛ Related Articles

1. Example 1: How many 3-digit numbers are there?

   Solution:

   There are 900 three-digit numbers in all. This can be calculated using the following method.

   • Step 1: Write the largest and the smallest 3-digit numbers. We know that the largest 3-digit number is 999. The smallest 3-digit number is 100.
   • Step 2: Find the difference between them. Their difference is, 999 - 100 = 899
   • Step 3: Add 1 to the difference. This means 899 + 1 = 900. Therefore, there are 900 three-digit numbers in all.

2. Example 2: Solve the puzzle: Add the smallest 2-digit number to the smallest 1-digit number. Subtract the sum from one less than the greatest 3-digit number.

   Solution:

   The smallest 2-digit number = 10. The smallest 1-digit number = 1. The sum of these two numbers is 10 + 1 = 11. One less than the greatest 3-digit number is 998. On subtracting 11 from 998, we get. 998 - 11 = 987.

3. Example 3: Find the greatest 3-digit number which is a perfect square.

   Solution: The greatest 3-digit number which is a perfect square is 961 because 312 = 961.

Show Solution >

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FAQs on Numbers up to 3-digits

There are a total of 900 three-digit numbers. These include the smallest 3 digit number - 100 to the largest 3 digit number - 999. The numbers beyond these 3-digit numbers are the 4-digit numbers, and the numbers less than the 3-digit numbers are 2-digit numbers.

Which is the Largest 3 Digit Number?

The largest 3-digit number is 999. Adding 1 more to it will make it a 4-digit number, that is, 1000.

What is the Sum of the Three Largest 3 Digit Numbers?

The three largest 3-digit numbers are 997, 998, 999. Their sum is 2994 as 997 +998 +999 = 2994.

What is the Smallest 3-Digit Number?

The number 100 is the smallest 3-digit number. Subtracting 1 from it makes it a 2-digit number. There are a total of 900 three-digit numbers, of which the number 100 is the smallest 3-digit number.

How Many Even 3-Digit Numbers are there?

There are a total of 900 3-digit numbers. Of these half of them are even numbers and the remaining half are odd numbers. Hence there are 900/2 = 450 even 3-digit numbers.

Can a 3-Digit Number have Two Zeros?

A 3-digit number can have two zeros. The two zeros should be in the tens place and the units place. Some of the examples of 3-digit numbers with two zeros are 100, 200, 300, and 400. It should be noted that the hundreds place in a 3-digit number cannot have the number 0 because that will make it a 2-digit number. For example, 098 becomes 98.

Which is the Smallest 3 Digit Number Divisible by 4?

The smallest 3-digit number is 100 and we know that it is divisible by 4 because 100/4 = 25. Therefore, we can say that 100 is the smallest 3-digit number which is divisible by 4.

Which 3 Digit Number has the Most Factors?

The 3-digit number that has the most factors is 840. The factors of 840 can be listed as, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420 and 840.