In a four‑digit number, the sum of the digits is 10. All the digits are different. What is the largest such four‑digit number?

Problems

Grades: 9th to 12th, 6th to 8th, 3rd to 5th

Mathematical Practices

Num & Ops Base Ten

Make sense of problems and persevere in solving them.

Generalize place value understanding for multi-digit whole numbers.

4.NBT.A.2, CCSS.Math.Practice.MP1

Water Bucket Conundrum
This problem has crossed my path a number of times in various guises. Perhaps you have also seen a version of it.
You are staying at a rural cabin, and the only method to get water is to draw it from a well. A 4-gallon bucket and a 9-gallon bucket are the only containers for carrying

Problems

Problems to Ponder: Neighborhood Sleuth

Problems

Place a number in each of the following empty boxes so that the sum
of the numbers in any 3 consecutive boxes is 2013. What is the number
that should go in the box with the question mark?

Problems

Grades: 9th to 12th, 6th to 8th, 3rd to 5th

Mathematical Practices

Num & Ops Base Ten

Reason abstractly and quantitatively.

Make sense of problems and persevere in solving them.

Use place value understanding and properties of operations to perform multi-digit arithmetic.

3.NBT.A.2, 4.NBT.B.4, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2

There are 4! = 24 ways to rank four objects. However, a friend told me that if ties
are allowed, the number increases to 75.

I
attempted to list all the possibilities by first listing the 24 orderings of
four objects, then using brackets to group ties involving two players, then
group ties involving three players, and finally the single case in which all
four objects are tied. But something has gone wrong; my list includes just 69
possibilities, not 75.

What
happened? Did I miss something, or was my friend mistaken?

Problems

Grades: 9th to 12th, 6th to 8th, 3rd to 5th

Mathematical Practices

Attend to precision.

Make sense of problems and persevere in solving them.

CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP6