Problem
Place number cards to make an equation true
There is one card each for 1, 2, 3, 4 and 5. Every card goes into exactly one of the five boxes. The sentence must come out true. Find three different fillings.
Reasoning · Grade 5-1 Number Puzzles
Your answer
Watch out Swapping cards looks new, but a swap inside the product is the same filling.
How to solve
Strategy Make a Systematic List — Only five small cards are in play, so the whole search is finite and a careful list will not miss anything. The trick that makes the list short is to look at what the two sides can be worth. The right side is at most 9, so the product on the left is at most 9 too - and there are only a handful of products of two different cards from 1 to 5 that are 9 or less. I list those products in increasing order, and for each one I check whether the three leftover cards can be arranged as (sum) divided by (card) to hit it.
1STEP 1
Find how big the right-hand side can be
The right side can never pass 9.
(4+5)/1=9 → right side ≤ 9
Reading the parentheses first is the whole skill here: the sum is computed before the division, so the largest sum over the smallest divisor gives the ceiling. Knowing the ceiling turns an open-ended puzzle into a short list.
5.OA.A.1Eliminate PossibilitiesBounding how large the right-hand side can be cuts the search down to a few products worth testing.
Why?
Bigger cards give bigger products, so once a product overshoots the bound every larger pairing overshoots too.
🧱Transitivity of orderIf the first is smaller than the second and the second smaller than the third, the first is smaller than the third, so pairwise comparisons chain into one full order.
Why?
Every pairing outside the bound is thrown out without arithmetic, so only a short list has to be tested by hand.
🧱Rule out the rest and the last one is forcedIf every possibility but one has been ruled out, the survivor must be the answer — no further checking is needed.
2STEP 2
List every product of two cards that is 9 or less
Products at most 9 are 2, 3, 4, 5, 6, 8.
1 × 2=2, 1 × 3=3, 1 × 4=4, 1 × 5=5, 2 × 3=6, 2 × 4=8
Listing products in order is just the times table you already know, and ordering them makes it obvious where to stop - once the products pass 9 they can never come back.
3.OA.C.7Make A Systematic List3STEP 3
Test each case with the three cards that are left over
Testing the leftovers leaves three that work.
(3+5)/4=2, (3+5)/2=4, (3+5)/1=8
Checking 'does this divide exactly' is the multiple-and-factor idea from Grade 4 - you only need to ask whether the sum is in the divisor's times table, which is a one-second check for numbers this small.
4.OA.B.4Guess And Check4STEP 4
Notice the pattern in the three winners
All three share 3 + 5 on the right.
(3+5)/4=2=1 × 2, (3+5)/2=4=1 × 4, (3+5)/1=8=2 × 4
Re-sorting the three answers by what they have in common shows the puzzle has one hidden hinge - the sum 8 - and that spotting the hinge would have produced all three at once.
5.OA.A.2Organize Information In More Ways5STEP 5
Confirm all five cards are used each time
Using all five cards gives the three with values 2, 4, 8.
The final check costs ten seconds and catches the most common slip in card puzzles, which is quietly reusing a card.
5.OA.A.1Make A Systematic ListAnswer
1 × 2 = (3 + 5) ÷ 4, 1 × 4 = (3 + 5) ÷ 2, 2 × 4 = (3 + 5) ÷ 1
Each way is checked by evaluating both sides separately: 1 x 2 = 2 and (3 + 5) / 4 = 8 / 4 = 2; 1 x 4 = 4 and (3 + 5) / 2 = 8 / 2 = 4; 2 x 4 = 8 and (3 + 5) / 1 = 8 / 1 = 8. All three divisions come out to whole numbers with nothing left over, and every value sits inside the range 1 to 9 that the cards allow. The list of six candidate products was complete, so there are no fourth or fifth ways hiding.
Takeaway
Ask how big each side can possibly get first - once you know the answer is at most 9, only six products survive and you can check them all.
- Find how big the right-hand side can be
- List every product of two cards that is 9 or less
- Test each case with the three cards that are left over
- Notice the pattern in the three winners
- Confirm all five cards are used each time