Return to Answer

Parameterize those numbers by $\overline {abc}$.

We have three cases, one when the first digit equals the sum of the other two. The second, when the second digit equals the sum of the other two, and so on...

Case I

$\overline {abc}$, $a=b+c$

when $b$ is $0$, $c$ can be $\{1,2,...,9\}$

when $b$ is $1$, $c$ can be $\{0,1,2,3,...,8\}$ and so on...

$9+9+8+...+1=55-1=54$ Case II

$\overline {bac}$, $a=b+c$

when $b$ is $1$, $c$ can be $\{0,1,...,8\}$

when $b$ is $2$, $c$ can be $\{0,1,...,7\}$ and so on...

Now, we've added some numbers here that we've also added in the previous case. Namely, the ones for which $b=a+c$ is true. Since we've added them in this case $b=a-c$ is also true, so $c$ must be $0$. $c$ is $0$ only when $b$ is $\{1,2,...,9\}$. So the final answer for this case is $45-9=36$. Case III

$\overline {bca}$, $a=b+c$

when $b$ is $1$, $c$ can be $\{0,1,...,8\}$

when $b$ is $2$, $c$ can be $\{0,1,...,7\}$ and so on..

Here we've added again some numbers, namely, for which $b$ is equal $a+c$, in step one. Note that we did not add any numbers that we added in case $2$ since, for those numbers we would have $c=b+a$(from case $2$) and also $c=a-b$(from step $3$). That would make $b$ $0$.(and we did not count that).

So, the final answer is $45-9=36$, for this case.

In total, we have $126$ numbers with that property.

Those numbers are: $$\begin{array}{|c|} \hline 101& 110& 112& 121& 123& 132& 134& 143& 145& 154& 156& 165& 167& 176& 178& 187&\\\hline 189& 198& 202& 211& 213& 220& 224& 231& 235& 242& 246& 253& 257& 264& 268& 275&\\\hline 279& 286& 297& 303& 312& 314& 321& 325& 330& 336& 341& 347& 352& 358& 363& 369&\\\hline 374& 385& 396& 404& 413& 415& 422& 426& 431& 437& 440& 448& 451& 459& 462& 473&\\\hline 484& 495& 505& 514& 516& 523& 527& 532& 538& 541& 549& 550& 561& 572& 583& 594&\\\hline 606& 615& 617& 624& 628& 633& 639& 642& 651& 660& 671& 682& 693& 707& 716& 718&\\\hline 725& 729& 734& 743& 752& 761& 770& 781& 792& 808& 817& 819& 826& 835& 844& 853&\\\hline 862& 871& 880& 891& 909& 918& 927& 936& 945& 954& 963& 972& 981& 990&\\\hline \end{array} $$ For verification, the code is:

 #include<iostream> using namespace std; int main() { int q=0; for(int a=1;a<=9;a++) for(int b=0;b<=9;b++) for(int c=0;c<=9;c++) if(a==b+c || b==a+c || c==a+b) {q++;cout<<a<<b<<c<<"\n";} cout<<q; return 0; } 

This, in case you do not allow $2-digit$ or $1-digit$ numbers.

Parameterize those numbers by $\overline {abc}$.

We have three cases, one when the first digit equals the sum of the other two. The second, when the second digit equals the sum of the other two, and so on...

Case I

$\overline {abc}$, $a=b+c$

when $b$ is $0$, $c$ can be $\{1,2,...,9\}$

when $b$ is $1$, $c$ can be $\{0,1,2,3,...,8\}$ and so on...

$9+9+8+...+1=55-1=54$

Case II

$\overline {bac}$, $a=b+c$

when $b$ is $1$, $c$ can be $\{0,1,...,8\}$

when $b$ is $2$, $c$ can be $\{0,1,...,7\}$ and so on...

Now, we've added some numbers here that we've also added in the previous case. Namely, the ones for which $b=a+c$ is true. Since we've added them in this case $b=a-c$ is also true, so $c$ must be $0$. $c$ is $0$ only when $b$ is $\{1,2,...,9\}$. So the final answer for this case is $45-9=36$.

Case III

$\overline {bca}$, $a=b+c$

when $b$ is $1$, $c$ can be $\{0,1,...,8\}$

when $b$ is $2$, $c$ can be $\{0,1,...,7\}$ and so on..

Here we've added again some numbers, namely, for which $b$ is equal $a+c$, in step one. Note that we did not add any numbers that we added in case $2$ since, for those numbers we would have $c=b+a$(from case $2$) and also $c=a-b$(from step $3$). That would make $b$ $0$.(and we did not count that).

So, the final answer is $45-9=36$, for this case.

In total, we have $126$ numbers with that property.

Those numbers are: $$\begin{array}{|c|} \hline 101& 110& 112& 121& 123& 132& 134& 143& 145& 154& 156& 165& 167& 176& 178& 187&\\\hline 189& 198& 202& 211& 213& 220& 224& 231& 235& 242& 246& 253& 257& 264& 268& 275&\\\hline 279& 286& 297& 303& 312& 314& 321& 325& 330& 336& 341& 347& 352& 358& 363& 369&\\\hline 374& 385& 396& 404& 413& 415& 422& 426& 431& 437& 440& 448& 451& 459& 462& 473&\\\hline 484& 495& 505& 514& 516& 523& 527& 532& 538& 541& 549& 550& 561& 572& 583& 594&\\\hline 606& 615& 617& 624& 628& 633& 639& 642& 651& 660& 671& 682& 693& 707& 716& 718&\\\hline 725& 729& 734& 743& 752& 761& 770& 781& 792& 808& 817& 819& 826& 835& 844& 853&\\\hline 862& 871& 880& 891& 909& 918& 927& 936& 945& 954& 963& 972& 981& 990&\\\hline \end{array} $$

For verification, the code is:

 #include<iostream> using namespace std; int main() { int q=0; for(int a=1;a<=9;a++) for(int b=0;b<=9;b++) for(int c=0;c<=9;c++) if(a==b+c || b==a+c || c==a+b) {q++;cout<<a<<b<<c<<"\n";} cout<<q; return 0; } 

This, in case you do not allow $2-digit$ or $1-digit$ numbers.

when $b$ is $0$, $c$ can be ${1,2,...,9}$

when $b$ is $1$, $c$ can be ${0,1,2,3,...,8}$

...

$9+9+8+...+1=55-1=54$

when $b$ is $1$, $c$ can be ${0,1,...,8}$

when $b$ is $2$, $c$ can be ${0,1,...,7}$

...

Now, we've added some numbers here that we've also added in the previous case. Namely, the ones for which b=a+c is true. Since we've added them in this case b=a-c is also true, so c must be 0. c is 0 only when b is {1,2,...,9}. So the final answer for this case is 45-9.

when $b$ is $1$, $c$ can be ${0,1,...,8}$

when $b$ is $2$, $c$ can be ${0,1,...,7}$

...

So, the final answer is $45-9$, for this case.

Those numbers are:101 110 112 121 123 132 134 143 145 154 156 165 167 176 178 187 189 198 202 211 213 220 224 231 235 242 246 253 257 264 268 275 279 286 297 303 312 314 321 325 330 336 341 347 352 358 363 369 374 385 396 404 413 415 422 426 431 437 440 448 451 459 462 473 484 495 505 514 516 523 527 532 538 541 549 550 561 572 583 594 606 615 617 624 628 633 639 642 651 660 671 682 693 707 716 718 725 729 734 743 752 761 770 781 792 808 817 819 826 835 844 853 862 871 880 891 909 918 927 936 945 954 963 972 981 990

Verify, we the code :

Case I

when $b$ is $0$, $c$ can be $\{1,2,...,9\}$

when $b$ is $1$, $c$ can be $\{0,1,2,3,...,8\}$ and so on...

$9+9+8+...+1=55-1=54$ Case II

when $b$ is $1$, $c$ can be $\{0,1,...,8\}$

when $b$ is $2$, $c$ can be $\{0,1,...,7\}$ and so on...

Now, we've added some numbers here that we've also added in the previous case. Namely, the ones for which $b=a+c$ is true. Since we've added them in this case $b=a-c$ is also true, so $c$ must be $0$. $c$ is $0$ only when $b$ is $\{1,2,...,9\}$. So the final answer for this case is $45-9=36$. Case III

when $b$ is $1$, $c$ can be $\{0,1,...,8\}$

when $b$ is $2$, $c$ can be $\{0,1,...,7\}$ and so on..

So, the final answer is $45-9=36$, for this case.

Those numbers are: $$\begin{array}{|c|} \hline 101& 110& 112& 121& 123& 132& 134& 143& 145& 154& 156& 165& 167& 176& 178& 187&\\\hline 189& 198& 202& 211& 213& 220& 224& 231& 235& 242& 246& 253& 257& 264& 268& 275&\\\hline 279& 286& 297& 303& 312& 314& 321& 325& 330& 336& 341& 347& 352& 358& 363& 369&\\\hline 374& 385& 396& 404& 413& 415& 422& 426& 431& 437& 440& 448& 451& 459& 462& 473&\\\hline 484& 495& 505& 514& 516& 523& 527& 532& 538& 541& 549& 550& 561& 572& 583& 594&\\\hline 606& 615& 617& 624& 628& 633& 639& 642& 651& 660& 671& 682& 693& 707& 716& 718&\\\hline 725& 729& 734& 743& 752& 761& 770& 781& 792& 808& 817& 819& 826& 835& 844& 853&\\\hline 862& 871& 880& 891& 909& 918& 927& 936& 945& 954& 963& 972& 981& 990&\\\hline \end{array} $$ For verification, the code is:

We have three cases, one when the first digit equals the sum of the other two. The second, when the second digit equals the sum of the other two...

when $b$ is $2$, $c$ can be ${0,1,...,7}$  ...

when $b$ is $2$, $c$ can be ${0,1,...,7}$ ...

$9+8+...+1=45$

Here we've added again some numbers, namely, for which b is equal a+c, in step one. Note that we did not add any numbers that we added in case 2 since, for those numbers we would have c=b+a(from case 2) and also c=a-b(from step 3). That would make b 0.(and we did not count that).

So, the final answer is 45-9, for this case.

In total, we have 126 numbers with that property.

Those numbers are:101 110 112 121 123 132 134 143 145 154 156 165 167 176 178 187 189 198 202 211 213 220 224 231 235 242 246 253 257 264 268 275 279 286 297 303 312 314 321 325 330 336 341 347 352 358 363 369 374 385 396 404 413 415 422 426 431 437 440 448 451 459 462 473 484 495 505 514 516 523 527 532 538 541 549 550 561 572 583 594 606 615 617 624 628 633 639 642 651 660 671 682 693 707 716 718 725 729 734 743 752 761 770 781 792 808 817 819 826 835 844 853 862 871 880 891 909 918 927 936 945 954 963 972 981 990 126

#include using namespace std;

int main() { int q=0;

for(int a=1;a<=9;a++) for(int b=0;b<=9;b++) for(int c=0;c<=9;c++) if(a==b+c || b==a+c || c==a+b) {q++;cout<<a<<b<<c<<"\n";}

cout<<q; return 0; }

This, in case you do not allow numbers of form $\overline{0bc}$.

We have three cases, one when the first digit equals the sum of the other two. The second, when the second digit equals the sum of the other two, and so on...

when $b$ is $2$, $c$ can be ${0,1,...,7}$ 

...

when $b$ is $2$, $c$ can be ${0,1,...,7}$

...

Here we've added again some numbers, namely, for which $b$ is equal $a+c$, in step one. Note that we did not add any numbers that we added in case $2$ since, for those numbers we would have $c=b+a$(from case $2$) and also $c=a-b$(from step $3$). That would make $b$ $0$.(and we did not count that).

So, the final answer is $45-9$, for this case.

In total, we have $126$ numbers with that property.

Those numbers are:101 110 112 121 123 132 134 143 145 154 156 165 167 176 178 187 189 198 202 211 213 220 224 231 235 242 246 253 257 264 268 275 279 286 297 303 312 314 321 325 330 336 341 347 352 358 363 369 374 385 396 404 413 415 422 426 431 437 440 448 451 459 462 473 484 495 505 514 516 523 527 532 538 541 549 550 561 572 583 594 606 615 617 624 628 633 639 642 651 660 671 682 693 707 716 718 725 729 734 743 752 761 770 781 792 808 817 819 826 835 844 853 862 871 880 891 909 918 927 936 945 954 963 972 981 990

 #include<iostream> using namespace std; int main() { int q=0; for(int a=1;a<=9;a++) for(int b=0;b<=9;b++) for(int c=0;c<=9;c++) if(a==b+c || b==a+c || c==a+b) {q++;cout<<a<<b<<c<<"\n";} cout<<q; return 0; } 

This, in case you do not allow $2-digit$ or $1-digit$ numbers.