Reasoning · Grade 2-1 Finding Numbers That Fit Conditions

Problem

Make the largest and smallest number

Four people each use their three number cards once to build a three-digit number. A three-digit number cannot begin with 0. Name who can make the overall largest number and who the smallest, in that order.
Your answer
How to solve
Strategy Identify Subproblems — Making the largest number means putting the biggest digit in the hundreds place and decreasing from there; making the smallest means the reverse, but never starting with 0. I work out each person's best and worst number, then compare the four results.
1STEP 1

Rule for largest and smallest

Largest goes biggest-first; smallest goes smallest-first but never leading with 0.

largest: d₁ ≥ d₂ ≥ d₃, smallest: d₃ ≤ d₂ ≤ d₁ (no leading 0)
2STEP 2

Each person's largest number

Each person's best is 540, 761, 930, 871.

540, 761, 930, 871
3STEP 3

Each person's smallest number

With 0 barred from the front, the smallests are 405, 167, 309, 178.

405, 167, 309, 178
4STEP 4

Compare to find the winners

Comparing, Noah's 930 is the largest and Mia's 167 the smallest.

max(540,761,930,871)=930, min(405,167,309,178)=167
Answer
Noah, Mia
930, 167
Noah's 930 starts with 9, the biggest hundreds digit anyone has, so it must be the largest overall. Mia's 167 starts with 1 and has the smaller tens digit (6) compared with Emma's 178 (tens 7), so 167 is the smallest overall. Liam's and Noah's smallest numbers each lose a 0 to the tens place, which keeps them above 167.
Takeaway

Big digits go on the left for the biggest number, small digits on the left for the smallest number, and never start with 0!

  • Rule for largest and smallest
  • Each person's largest number
  • Each person's smallest number
  • Compare to find the winners