# Make expressions equal to 100 using exactly seven 4s

Use exactly seven 4s in every expression and no other digits/numbers.

• Choose from among addition, subtraction, division, and/or multiplication operations.

• You may use parentheses, brackets, and/or braces for grouping and/or multiplication, as needed.

• This is in base ten. (The numbers using the fours and 100 on the other side of the equals sign are in base 10.)

• You may not use decimal points.

• No concatenation is allowed.

• You may not use factorial signs.

• You may not use square roots.

• You may not use exponentiation.

• No other characters or operations may be used.

Note: Expressions with forms of A + B and B + A will be considered the same, as will be -A + B with B - A, and A(B) with B(A), and A(B) + C with C + A(B), etc.

Create a minimum of six essentially different solutions based on the note above.

• Somewhat related I guess. May the fours be with you.
– CG.
Jan 30, 2020 at 9:07
• Given that concatenation is forbidden, how can "the numbers using the fours" even have a base? Jan 31, 2020 at 1:21
• Since you want a minimum of xx different solutions, why the restriction that concatenation is not allowed? Just accept 44+44+4+4+4 and demand at least 7 different solutions. Jan 31, 2020 at 8:24
• @ Mr. Lister - With concatenation of digits, it 1) makes this puzzle less challenging, and 2) I am trying to keep the total number of different solutions down because including concatenation would add at least three more solutions. Feb 1, 2020 at 14:57

My suggestion:

$$100=4\cdot(4\cdot4+4+4+4\div4)$$
$$100=4\cdot(4+4\div4)\cdot(4+4\div4)$$
$$100=4\cdot(4\cdot(4+4\div4)+4)+4$$
$$100=4\cdot(4\cdot(4+4)-4-4)+4$$
$$100= 4\cdot(4+4)\cdot(4-4\div4)+4$$
$$100=4\cdot4\cdot4+4\cdot(4+4)+4$$

• That was so fast after I posted! Jan 29, 2020 at 22:51

Different to @ThomasL's solution, I found:

$$(4\times4)(4+\frac{4+4}{4})+4$$

• @ JMP - That was pretty fast after I posted the puzzle. Jan 29, 2020 at 22:57
• There just aren't that many options :)
– JMP
Jan 29, 2020 at 23:47

Here is a newer one of my own:

(4*4 + 4)(4*4 + 4)/4 = 100

(4*4 + 4)*4 + 4*4 + 4
= 20 * 4 + 16 + 4

• @ SOLO - I looked at this about two hours after you posted it. Thank you for it. Jan 30, 2020 at 2:58

The phrase

This is in base ten. (The numbers using the fours and 100 on the other side of the equals sign are in base 10.)

makes me think using 44 is allowed. If that's the case then:

44 + 44 + 4 + 4 + 4 = 100

• But concatenation is explicitly disallowed. Jan 31, 2020 at 1:13
• I thought that concatenation is explicitly disallowed as its own operator. (like using it on the result of another expression) example: ((4 + 4) ^ 4) + 4 + 4 + 4 + 4 where ^ denotes concatenation.
– pufe
Jan 31, 2020 at 17:26
• @RossMillikan if "no concatenation is allowed" means that you can't use 44, then there's no reason to say "The numbers using the fours...are in base 10.". The digit 4 has the same value in all bases except for those bases that do not include the digit 4. Jan 31, 2020 at 19:07

You could do:

$$((\frac{4}{4} + 4)\times 4 \times (\frac{4}{4} + 4) =100 )$$

• This is the same (up to reordering of the multiplication) as the second line in ThomasL's accepted solution. Jan 31, 2020 at 13:59