Geometry Puzzle – Critical Thinking 20
How many three-digit numbers have digits that add to 7? It looks like a counting question, but it's really a clean stars-and-bars problem with one restriction: the hundreds digit must be at least 1. Once you spot the structure, the answer falls out fast.
Key takeaways:
- A three-digit number \( \overline{abc} \) has hundreds digit \( a \in \{1, \ldots, 9\} \) and \( b, c \in \{0, \ldots, 9\} \).
- We want \( a + b + c = 7 \) with \( a \geq 1 \).
- Substitute \( a-prime= a - 1 \) (so \( a-prime\geq 0 \)): the equation becomes \( a-prime+ b + c = 6 \).
- Count nonnegative integer solutions: \( \binom{6+2}{2} = \binom{8}{2} = 28 \).
- Since each digit is at most 7 (which is at most 9), no upper-bound constraint kicks in, the answer is 28.
To solve this kind of challenge, sometimes it just needs logical steps and other times it involves lateral out-of-the-box thinking!
Challenge:
In how many 3-digit numbers the sum of all digits is 7?
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The correct answer is 28.
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First, let’s find all three digits that add up to 7.
700, 610, 511, 520, 430, 421, 322, 133
For 7, 0 and 0, we can write only one three digit number which is 700.
For 6, 1 and 0, we can find 106, 160, 601, 610,
For 5, 1, and 1: 115, 151, 511
For 5, 2, and 0: 205, 250, 502, 520
For 4, 3, and 0: 304, 340, 403, 430
For 4, 2, and 1: 124, 142, 214, 241, 412, 421
For 3, 2, and 2: 223, 232, 322
For 1, 3, and 3: 133, 313, 331
There are 28 three digits numbers that the sum of their digits is 7.
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Frequently Asked Questions
Is math critical thinking?
Math is one of the best ways to practice critical thinking, because you have to decide what a problem is really asking before you calculate. In this puzzle, the key insight is that the hundreds digit cannot be 0. Once you set a = a′ + 1, counting three-digit numbers whose digits add to 7 becomes counting solutions to a′ + b + c = 6, which gives 28.
How do you solve math puzzles step by step?
Read the puzzle carefully, list the rules, try a few small cases, look for a pattern, then confirm with a second method. For three-digit numbers with digit sum 7, fix the hundreds digit a from 1 to 7. Each choice leaves b + c = 7 − a, which has 8 − a pairs. Adding 7 + 6 + 5 + 4 + 3 + 2 + 1 gives 28.
Are puzzles critical thinking?
Yes. Good puzzles make you question your first approach, organize information and check your answer, which is exactly what critical thinking means. Listing every number whose digits add to 7 works but is easy to get wrong. Grouping them by digit sets, like {7, 0, 0} giving only 700 and {4, 2, 1} giving six arrangements, keeps the count of 28 organized and easy to check.
Are there critical thinking math puzzles in PDF?
Yes, many critical thinking math puzzles are available as printable PDFs, and this digit-sum challenge is easy to print and try on paper too. Working by hand is useful here: write the digit groups that add to 7, such as 6 + 1 + 0 and 5 + 2 + 0, then count every way to arrange each group without a leading zero.
Is math just puzzles?
Not just puzzles, but a lot of math feels like one. Math also includes facts, procedures and real-world modeling, and puzzles connect those pieces. This one starts as a counting question about digits and turns into combinatorics: the stars-and-bars formula C(8, 2) = 28 is the same tool used to count ways to share identical items among people.
Related Lessons You May Like
- How to solve permutations and combinations
- How to find the probability of an event
- How to solve multi-step word problems
- How to add and subtract multi-digit numbers
- Place value up to thousands
If your student likes puzzles like this one, Geometry for Beginners works the same kinds of relationships inside a full geometry curriculum. For the algebra you’ll lean on for setting up the equations, Pre-Algebra for Beginners fills in the foundations gently.
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