The 5-4-3-2-1 Challenge | Math = Love - Free Printable

I need to solve the 5-4-3-2-1 Challenge. The task is to use the digits 5, 4, 3, 2, and 1 in that exact order, and place arithmetic operations (+, -, ×, ÷) between them, along with parentheses as needed, to create expressions that equal numbers from 1 to 40.

Let me work through this systematically to find which numbers from 1 to 40 can be created.

I'll start by finding expressions for each number:

Number 1:
- 5 - 4 + 3 - 2 - 1 = 1 ✓

Number 2:
- 5 - 4 + 3 - 2 × 1 = 2? Let's check: 5 - 4 + 3 - 2 = 2 ✓
- Or: (5 + 4) ÷ 3 + 2 - 1 = 9 3 + 2 - 1 = 3 + 2 - 1 = 4 ✗
- Try: 5 - 4 + 3 ÷ 3 × 1... no
- 5 + 4 - 3 - 2 - 1 = 8 - 6 = 2 ✓

Number 3:
- 5 - 4 + 3 + 2 - 1 = 5 ✓ No
- 5 + 4 - 3 - 2 - 1 = 3 ✓

Number 4:
- 5 + 4 - 3 - 2 × 1 = 4? 5 + 4 - 3 - 2 = 4 ✓
- Or: (5 + 4 + 3) 2 - 1 = 12 ÷ 2 - 1 = 6 - 1 = 5 ✗
- (5 - 4) × (3 + 2 - 1) = 1 × 4 = 4 ✓

Number 5:
- 5 + 4 - 3 - 2 + 1 = 5 ✓

Number 6:
- 5 + 4 - 3 × (2 - 1) = 5 + 4 - 3 = 6 ✓
- Or: 5 + 4 - 3 + 2 - 1 = 7 ✗
- 5 - 4 + 3 + 2 × 1 = 6 ✓

Number 7:
- 5 + 4 - 3 + 2 - 1 = 7 ✓

Number 8:
- 5 + 4 - 3 + 2 × 1 = 8? 5 + 4 - 3 + 2 = 8 ✓

Number 9:
- 5 + 4 + 3 - 2 - 1 = 9 ✓

Number 10:
- 5 + 4 + 3 - 2 × 1 = 10? 5 + 4 + 3 - 2 = 10 ✓

Number 11:
- 5 + 4 + 3 - 2 + 1 = 11 ✓

Number 12:
- 5 + 4 + 3 × (2 - 1) = 5 + 4 + 3 = 12 ✓
- Or: (5 + 4 + 3) × (2 - 1) = 12 ✓

Number 13:
- 5 + 4 + 3 + 2 - 1 = 13 ✓

Number 14:
- 5 + 4 + 3 + 2 × 1 = 14? 5 + 4 + 3 + 2 = 14 ✓

Number 15:
- 5 + 4 + 3 + 2 + 1 = 15 ✓

Number 16:
- 5 × 4 - 3 - 2 + 1 = 20 - 4 = 16 ✓

Number 17:
- 5 × 4 - 3 - 2 × 1 = 20 - 3 - 2 = 15 ✗
- 5 × 4 - 3 × (2 - 1) = 20 - 3 = 17 ✓

Number 18:
- 5 × 4 - 3 + 2 - 1 = 20 - 2 = 18 ✓

Number 19:
- 5 × 4 - 3 + 2 × 1 = 20 - 3 + 2 = 19 ✓

Number 20:
- 5 × 4 × (3 - 2) × 1 = 20 ✓
- Or: 5 × 4 + 3 - 2 - 1 = 20 ✓

Number 21:
- 5 × 4 + 3 - 2 × 1 = 20 + 3 - 2 = 21 ✓

Number 22:
- 5 × 4 + 3 - 2 + 1 = 22 ✓

Number 23:
- 5 × 4 + 3 × (2 - 1) = 20 + 3 = 23 ✓

Number 24:
- 5 × 4 + 3 + 2 - 1 = 24 ✓

Number 25:
- 5 × 4 + 3 + 2 × 1 = 25? 20 + 3 + 2 = 25 ✓

Number 26:
- 5 × 4 + 3 + 2 + 1 = 26 ✓

Number 27:
- 5 × 4 + 3 × 2 - 1 = 20 + 6 - 1 = 25 ✗
- (5 + 4) × 3 × (2 - 1) = 9 × 3 = 27 ✓

Number 28:
- 5 × 4 + 3 × 2 × 1 = 20 + 6 = 26 ✗
- (5 + 4) × 3 + 2 - 1 = 27 + 2 - 1 = 28 ✓

Number 29:
- (5 + 4) × 3 + 2 × 1 = 27 + 2 = 29 ✓

Number 30:
- (5 + 4) × 3 + 2 + 1 = 30 ✓
- Or: 5 × (4 + 3 - 2 + 1) = 5 × 6 = 30 ✓

Number 31:
- 5 × (4 + 3) - 2 - 1 = 35 - 3 = 32 ✗
- (5 × 4 + 3) × 2 - 1 = 23 × 2 - 1 = 45 ✗
- 5 × 4 + 3 × 2 + 1 = 20 + 6 + 1 = 27 ✗
- (5 + 4 × 3) × 2 - 1 = 17 × 2 - 1 = 33 ✗
- Let me try: 5 × (4 + 3) - 2 × 1 = 35 - 2 = 33 ✗
- Hmm, let me think differently: (5 × 4 - 3) × 2 - 1 = 17 × 2 - 1 = 33 ✗
- 5 × 4 × 2 - 3 + 1 = 40 - 2 = 38 ✗ (wrong order)
- Actually wait, I need to keep the order 5-4-3-2-1
- Let me try: ((5 + 4) × 3 - 2) × 1 = 25
- (5 × 4 + 3 + 2) × 1 = 25 ✗
- 5 × (4 + 3 - 2) + 1 = 5 × 5 + 1 = 26 ✗
- 5 × (4 + 3) - (2 - 1) = 35 - 1 = 34 ✗
- This one is tricky. Let me skip it for now.

Number 32:
- 5 × (4 + 3) - 2 - 1 = 35 - 3 = 32 ✓

Number 33:
- 5 × (4 + 3) - 2 × 1 = 35 - 2 = 33 ✓

Number 34:
- 5 × (4 + 3) - (2 - 1) = 35 - 1 = 34 ✓

Number 35:
- 5 × (4 + 3) × (2 - 1) = 35 ✓
- Or: 5 × 4 + 3 × 2 × 1... wait that's 26
- 5 × (4 + 3) + 2 - 1 = 35 + 1 = 36
- 5 × (4 + 3) × 2 - 1 would require different grouping
- Actually: (5 × 4 - 3) × 2 + 1 = 17 × 2 + 1 = 35 ✓

Number 36:
- 5 × (4 + 3) + 2 - 1 = 36 ✓

Number 37:
- 5 × (4 + 3) + 2 × 1 = 37 ✓

Number 38:
- 5 × (4 + 3) + 2 + 1 = 38 ✓

Number 39:
- (5 + 4) × (3 + 2) - 1 = 9 × 5 - 1 = 44
- 5 × 4 × 2 - 3 + 1... wrong order
- (5 × 4 + 3) × 2 - 1 = 23 × 2 - 1 = 45 ✗
- Hmm, let me try: (5 + 4) × 3 + 2 × 1 = 29 ✗
- This might not be possible. Let me continue checking.

Number 40:
- 5 × 4 × (3 - 2 + 1) = 5 × 4 × 2 = 40 ✓
- Or: (5 + 4 + 3 - 2) × 1... no that's 10
- 5 × 4 × 2 × 1... but I need to use 3
- (5 × 4) × (3 - 2) × 1 = 20
- Wait: 5 × 4 × (3 - 2 + 1) = 5 × 4 × 2 = 40 ✓

Now let me go back and check which numbers I couldn't find. I had trouble with 21 (but found it), 31, and 39.

Let me reconsider 31:
- Need to get 31 from 5-4-3-2-1
- 5 × 4 + 3 × 2 + 1 = 20 + 6 + 1 = 27 ✗
- (5 + 4) × 3 + 2 + 1 = 30
- 5 × (4 + 3) - 2 - 1 = 32
- What about: (5 × 4 - 3) × 2 - 1 = 17 × 2 - 1 = 33 ✗
- Or: 5 × 4 + (3 + 2) × 1 = 20 + 5 = 25 ✗
- (5 + 4 × 3) × 2 - 1 = 17 × 2 - 1 = 33
- 5 × (4 + 3 - 2) + 1 = 25 + 1 = 26 ✗
- Hmm, maybe 31 isn't possible?

Let me try 39 again:
- (5 + 4) × (3 + 2) - 1 = 45 - 1 = 44 ✗
- 5 × 4 × 2 - 3 + 1... can't do that with the order
- (5 × 4 + 3) × 2 - 1 = 45 ✗
- 5 × (4 + 3) + 2 × 1 = 37
- Maybe 39 isn't possible either?

Let me verify some of my earlier answers more carefully and see if I missed any numbers or made errors.

Actually, looking at the worksheet in the image, it shows a table going up to at least 28, suggesting many numbers are achievable. Let me be more systematic and verify each number I found, then identify which ones from 1-40 cannot be achieved.

After careful analysis, most numbers from 1-40 can be achieved. The ones that appear to be impossible (or very difficult) are typically 31 and 39, though I should verify this more carefully.

Based on my calculations, I can achieve: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40

That's 38 numbers. The missing ones appear to be 31 and 39.

However, let me double-check 31 and 39 one more time:

For 31:
- Trying various combinations with proper order 5-4-3-2-1
- I cannot find a valid expression that equals 31

For 39:
- Similarly, I cannot find a valid expression that equals 39

So the answer is that you can get 38 numbers from 1 to 40.

Final Answer: 38