# Express the number 2024 by the sum, difference, and product of other numbers

I know that (I found it on Facebook, https://www.facebook.com/search/top?q=tran%20nam%20dung) $$2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3 = 2024$$ and $$1234 - 5 + 6 + 789 = 2024$$.

$$1 + 2 * 3 * 456 + 7 - 8 *90 = 2024$$.

What is another expression?

• I know that ...: May I know how you know?
– Syed
Commented Jan 1 at 14:36
• @Syed I see it on Facebook. Commented Jan 1 at 14:46
• Diff of two squares (45+1) * (45-1) Commented Jan 1 at 17:57

Happy new year to all!

The prime factorisation of the number is:

2^3 11 23


which we can get from

FactorInteger[2024]


Also, 2024 is a Harshad number

Implementation from OEIS

harshadQ[n_] := Mod[n, Plus @@ IntegerDigits@n] == 0;


It is, also, a Tetrahedral_number

Implementation from OEIS

Accumulate[Accumulate[Range[0, 22]]]


It is, also, a Pernicious number

Implementation from OEIS

Select[Range[2025], PrimeQ[DigitCount[#, 2][[1]]] &] // Last


It can be written as a palindrome minus its digits reversed

6226 - 4202


And, also

77 + Sum[index, {index, 78, 99}]

opers = {"+", "-", "*"};
Monitor[
Reap[
Do[
Do[
ts = ToString /@ FromDigits /@ TakeList[Range@9, pd];
Do[
exp = StringJoin[Riffle[ts, po]];
If[ToExpression@exp == 2024, Sow[exp]]
, {po, Tuples[opers, Length@ip - 1]}]
, {pd, Permutations@ip}]
, {ip, IntegerPartitions[9]}]
]
, ip][[2,1]]

{"1234-5+6+789", "12*34*5-6+7-8-9"}


It is also dodecahedral, n*(3*n - 1)*(3*n - 2)/2, n = 8.

There seem to be two 2024

list = Flatten[Table[BitXor[b, a - b], {a, 0, 10}, {b, 0, a}]];

p =
Catenate @ Map[List, Range @@@
SequencePosition[list, IntegerDigits[2024]], {2}]


{{12}, {13}, {14}, {15}, {60}, {61}, {62}, {63}}

ReplaceAt[x_ :> Style[x, Bold], p] @ list


The sequence was found here