Reasoning · Grade 2-1 Finding Numbers That Fit Conditions

Problem

Deduce the hidden flipped cards

Of three cards two show 6 and 3, and the third is face down. The hidden digit is 1 to 9 and is neither 6 nor 3. Using all three cards once to build three-digit numbers, find the largest and smallest for each case: hidden below 3, between 3 and 6, and above 6.
Your answer
How to solve
Strategy Make a Systematic List — The hidden digit has only a few possible values in each case, so I list them all. For each set of three digits, the largest number sorts the digits from biggest to smallest and the smallest sorts them from smallest to biggest.
1STEP 1

Rule for largest and smallest from three digits

With the digits fixed, biggest-first is the largest, smallest-first the smallest.

largest=d_bigd_midd_small, smallest=d_smalld_midd_big
2STEP 2

Case 1: hidden digit less than 3 (hidden = 1 or 2)

Below 3 the hidden card is 1 or 2, giving 631/136 and 632/236.

□=1: 631, 136; □=2: 632, 236
3STEP 3

Case 2: hidden digit between 3 and 6 (hidden = 4 or 5)

Between 3 and 6 it is 4 or 5, giving 643/346 and 653/356.

□=4: 643, 346; □=5: 653, 356
4STEP 4

Case 3: hidden digit greater than 6 (hidden = 7, 8, or 9)

Above 6 it is 7, 8 or 9, giving 763/367, 863/368, 963/369.

□=7: 763, 367; □=8: 863, 368; □=9: 963, 369
Answer
1→631/136, 2→632/236, 4→643/346, 5→653/356, 7→763/367, 8→863/368, 9→963/369
In every case the three digits are distinct and nonzero, so each largest number sorts them downward and each smallest sorts them upward. Spot check square=5: digits 6,5,3 give 653 (down) and 356 (up), both correct. When the hidden digit is below 6, the hundreds digit of the largest number stays 6; when above 6, the hidden digit takes the hundreds place, which matches the rising answers 763, 863, 963.
Takeaway

List the few values the hidden card can be, then for each just sort the three digits down for the biggest number and up for the smallest!

  • Rule for largest and smallest from three digits
  • Case 1: hidden digit less than 3 (hidden = 1 or 2)
  • Case 2: hidden digit between 3 and 6 (hidden = 4 or 5)
  • Case 3: hidden digit greater than 6 (hidden = 7, 8, or 9)